Gr 10 Math June P2 Hard

What can be inferred about the lengths of corresponding sides in similar triangles?

They are always equal regardless of the triangle's size.

They are unrelated to the angles of the triangles.

They form the basis of non-Euclidean geometry.

They are proportional but may differ in length. (correct)

Which of the following statements about the angles in similar triangles is true?

Corresponding angles must be equal for the triangles to be similar. (correct)

The triangles can be similar even if one angle differs.

The angles can be different but still result in proportional sides.

All angles must be 90 degrees for the triangles to be similar.

How does the order of the vertices affect the similarity of two triangles?

It determines which sides are proportional to each other. (correct)

Changing the order alters the ratio of corresponding sides.

The order does not affect the similarity as long as the angles are equal.

It has no bearing; only the areas of the triangles matter.

What is a key characteristic of the ratios of corresponding sides in similar triangles?

They remain constant irrespective of the angles. Signup and view all the answers

In the case of triangles $ riangle ABC$, $ riangle DEF$, and $ riangle GHK$, which ratio is expected to be consistent?

$rac{AB}{DE}$ Signup and view all the answers

Which statement correctly describes the concept of trigonometric ratios?

They are derived from the side ratios of similar triangles. Signup and view all the answers

What is the fundamental requirement for triangles to be considered similar?

All corresponding angles must be equal. Signup and view all the answers

Which of the following ratios illustrates the relationship between similar triangles accurately?

$rac{GH}{HK} = rac{AC}{DE}$ Signup and view all the answers

What effect does an amplitude change with property |a| > 1 have on the cosine function?

It vertically stretches the graph. Signup and view all the answers

Which classification of triangle describes a triangle with all sides of different lengths?

Scalene Signup and view all the answers

When comparing the sine and cosine functions, how can the cosine graph be visually aligned with the sine graph?

By shifting the cosine graph 90° to the right. Signup and view all the answers

What is the range of the function y = a cos θ + q when a > 0?

[q - |a|, q + |a|] Signup and view all the answers

For the tangent function y = a tan θ + q, what does the variable q represent?

The vertical shift of the graph. Signup and view all the answers

Which statement about a triangle's exterior angles is correct?

An exterior angle is equal to the sum of the two adjacent interior angles. Signup and view all the answers

Which rule would be used to prove two triangles congruent if two sides and the included angle are equal?

SAS Signup and view all the answers

In the context of triangle similarity, what does the SSS criterion specifically require?

All three sides must be in proportion. Signup and view all the answers

Which of the following describes the y-intercept of the function y = a tan θ + q?

y = a tan 0° + q = q Signup and view all the answers

What is the period of the tangent function y = tan θ?

180° Signup and view all the answers

Which of the following statements is not a property of a parallelogram?

All angles are equal. Signup and view all the answers

What can be concluded if a quadrilateral is identified as a kite?

It has diagonals that are perpendicular. Signup and view all the answers

What is the relationship between a square and a rectangle?

A square is a specific type of rectangle. Signup and view all the answers

What is the value of the sine function at an angle of 45°?

$\frac{\sqrt{2}}{2}$ Signup and view all the answers

In a 30°-60°-90° triangle, what is the ratio of the length of the hypotenuse to the length of the side opposite the 30° angle?

2:1 Signup and view all the answers

According to the Mid-Point Theorem, what is true about the segment connecting the mid-points of two sides of a triangle?

It is parallel to the triangle's base. Signup and view all the answers

What is the tangent of angle θ if the opposite side is 4 and the adjacent side is 2?

2 Signup and view all the answers

Which property is unique to a rhombus among quadrilaterals?

All sides are of equal length. Signup and view all the answers

If a triangle has sides measuring 3, 4, and 5, what can be deduced about this triangle?

It is a right-angled triangle. Signup and view all the answers

Which of the following angles has a cosine value of 1/2?

60° Signup and view all the answers

When using the inverse sine function, what is the valid range of outputs?

$[-90°, 90°]$ Signup and view all the answers

What characterizes a trapezium in relation to its sides?

It has at least one pair of opposite sides parallel. Signup and view all the answers

If $\tan B = \frac{3}{2}$ for point Q(-2, 3), what is the value of angle B?

123.7° Signup and view all the answers

How do the angles of a quadrilateral compare to each other?

They must add up to 360°. Signup and view all the answers

Which statement is true regarding the diagonals of a rectangle?

They are equal in length. Signup and view all the answers

What is the relationship between the sides of a 45°-45°-90° triangle?

Both legs are equal and the hypotenuse is $\sqrt{2}$ times the length of each leg. Signup and view all the answers

What restrictions apply to the values of sine and cosine functions?

$|\sin θ| \leq 1$ and $|\cos θ| \leq 1$ Signup and view all the answers

In the context of trigonometric ratios, what does the tangent function represent?

The ratio of the opposite side to the adjacent side. Signup and view all the answers

What is the gradient of a line that runs parallel to the x-axis?

0 Signup and view all the answers

What can be said about the gradients of two perpendicular lines?

They are negative inverses of each other. Signup and view all the answers

To find the midpoint of a line segment between points A(4, 2) and B(6, 8), what are the coordinates of the midpoint?

(5, 6) Signup and view all the answers

What is the slope formula used for determining the gradient of a line between two points?

$rac{y₂ - y₁}{x₂ - x₁}$ Signup and view all the answers

If line WX is perpendicular to line YZ and the gradient of line WX is 2, what is the gradient of line YZ?

-0.5 Signup and view all the answers

Given points A(2, 3) and B(2, 5), what type of line do these points define?

Vertical line Signup and view all the answers

Which of the following statements about gradients is true?

Parallel lines have equal gradients. Signup and view all the answers

Which equation represents the standard form of a straight line?

$y = mx + c$ Signup and view all the answers

What is the value of $\tan 60^\circ$?

$\sqrt{3}$ Signup and view all the answers

In which quadrant is $\sin \theta$ positive and $\tan \theta$ negative?

Quadrant II Signup and view all the answers

If $a = -2$ and $q = 3$, what is the range of the function $y = a \sin \theta + q$?

$[1, 5]$ Signup and view all the answers

What is the period of the function $y = \cos \theta$?

360° Signup and view all the answers

Which of the following values corresponds to $\cos 30^\circ$?

$\frac{\sqrt{3}}{2}$ Signup and view all the answers

What is the y-intercept of the function $y = -3 \sin \theta + 2$?

$2$ Signup and view all the answers

Which trigonometric ratio is defined as $\frac{\text{adjacent}}{\text{hypotenuse}}$?

$\cos \theta$ Signup and view all the answers

What is the correct definition of the tangent ratio in a right-angled triangle?

$ an heta = rac{ ext{opposite}}{ ext{adjacent}}$ Signup and view all the answers

Under which condition is the value of $\tan \theta$ undefined?

When $\theta = 90^\circ$ Signup and view all the answers

Which of the following statements about the trigonometric ratios is true?

$ an heta = 1$ when $ heta = 45^ ext{o}$ Signup and view all the answers

What is the reciprocal of the cosine ratio?

secant Signup and view all the answers

What is the angle of elevation equal to when viewed from the same horizontal plane down to the ground?

The angle of depression Signup and view all the answers

When converting $ ext{cosec } heta$ to a standard trigonometric ratio, which expression is used?

$rac{ ext{hypotenuse}}{ ext{opposite}}$ Signup and view all the answers

Which pair of trigonometric ratios is reciprocally related?

sin and cosec Signup and view all the answers

For angle $ heta$, what does the relationship $ ext{sin } heta imes ext{cosec } heta = 1$ imply?

Sine and cosecant are multiplicative inverses. Signup and view all the answers

If a calculator is used to find $ ext{sec } 34^ ext{o}$, which standard trigonometric ratio should be used?

$rac{1}{ ext{cos } 34^ ext{o}}$ Signup and view all the answers

Which angle does not require a calculator to determine its trigonometric ratio?

$45^ ext{o}$ Signup and view all the answers

Which relation holds true among the trigonometric identities?

$ ext{sin }^2 heta + ext{cos }^2 heta = 1$ Signup and view all the answers

When employing the mnemonic 'Soh Cah Toa', what concept does 'Cah' represent?

$ an heta = rac{ ext{adjacent}}{ ext{hypotenuse}}$ Signup and view all the answers

What does the line segment joining the mid-points of two sides of a triangle represent in relation to the third side?

It is parallel to the third side and half its length. Signup and view all the answers

Which property is NOT used to prove that MNOP is a parallelogram?

Diagonals bisect each other. Signup and view all the answers

In the proof for parallelogram MNOP, which pair of triangles are shown to be congruent using the AAS criterion?

△CDZ and △ABX Signup and view all the answers

How is the distance formula derived for calculating the distance between two points in a Cartesian plane?

By applying the Pythagorean theorem to the changes in x and y. Signup and view all the answers

What characterizes the gradient of a line between two points?

It is the ratio of vertical change to horizontal change. Signup and view all the answers

Which feature of a quadrilateral must be proven to classify it as a parallelogram?

At least one pair of opposite sides are equal and parallel. Signup and view all the answers

What is the necessary conclusion about the angles in parallelogram MNOP?

Opposite angles are equal. Signup and view all the answers

To find the gradient of the line between two points A(x₁, y₁) and B(x₂, y₂), which values are used?

The differences in y and x values. Signup and view all the answers

Which of the following would NOT be a valid application of the Mid-Point Theorem?

Determining lengths in non-triangular shapes. Signup and view all the answers

When plotting a quadrilateral ABCD on the Cartesian plane, what is crucial about the order of vertices?

The sequence of letters must reflect the connection order. Signup and view all the answers

When comparing the ratios of corresponding sides of two similar triangles, what will be true regardless of the triangles' sizes?

The ratios will be equal. Signup and view all the answers

Which of the following statements accurately describes the importance of corresponding angles in similar triangles?

They must be equal for the triangles to be considered similar. Signup and view all the answers

If triangles $Δ ABC$ and $Δ DEF$ are similar with a given proportion of the sides, how would you express the relationship of the sides mathematically?

$rac{AB}{DE} = rac{BC}{EF} = rac{AC}{DF}$ Signup and view all the answers

What must be true about the order of vertices when determining triangle similarity?

Corresponding vertices must align based on the triangle's angles. Signup and view all the answers

What is the significance of calculating corresponding side ratios in similar triangles?

It allows for the definition of trigonometric ratios. Signup and view all the answers

In similar triangles, if the angles are known to be 30°, 60°, and 90°, how do the corresponding sides relate?

The ratio of the corresponding sides remains constant regardless of triangle size. Signup and view all the answers

Which of the following describes a common misconception about the relationship between angles and side lengths in similar triangles?

All corresponding angles must be different for the triangles to be similar. Signup and view all the answers

How do the properties of similar triangles relate to the study of trigonometry?

They provide foundational principles that help define trigonometric ratios. Signup and view all the answers

What is the formula for cosine in relation to a right-angled triangle?

$rac{ ext{adjacent}}{ ext{hypotenuse}}$ Signup and view all the answers

Which of the following relationships is true for the trigonometric ratios?

$ an heta imes ext{cot } heta = 1$ Signup and view all the answers

If the sine of an angle is $rac{3}{5}$, what is the value of cosecant at that angle?

$rac{5}{3}$ Signup and view all the answers

Given $ an heta = rac{9}{4}$, which of the following represents the corresponding sides of the triangle?

Opposite = 9, Adjacent = 4 Signup and view all the answers

What is the reciprocal of the tangent function?

Cotangent Signup and view all the answers

For what angle does $ an heta$ equal 0?

0° Signup and view all the answers

What must be true for a right triangle's sides in relation to the Pythagorean theorem?

$ ext{opposite}^2 + ext{adjacent}^2 = ext{hypotenuse}^2$ Signup and view all the answers

Which mnemonic is commonly used to remember trigonometric ratios?

Soh Cah Toa Signup and view all the answers

If the cosecant of an angle is $rac{7}{2}$, what is the sine of that angle?

$rac{2}{7}$ Signup and view all the answers

From the trigonometric ratios, if the adjacent side measures 5 and the hypotenuse measures 13, what is $rac{ ext{adjacent}}{ ext{hypotenuse}}$?

$rac{5}{13}$ Signup and view all the answers

What is a property that distinguishes a rhombus from a rectangle?

All sides are equal in length. Signup and view all the answers

In which of the following quadrilaterals are the diagonals guaranteed to bisect each other at right angles?

Kite Signup and view all the answers

What conclusion can be drawn about a quadrilateral if it has one pair of opposite sides equal and parallel?

It is a trapezium. Signup and view all the answers

Which statement correctly describes the relationship between a parallelogram and a rectangle?

All rectangles are parallelograms. Signup and view all the answers

If a triangle's mid-point theorem is applied, what relationship does the segment joining the mid-points of the triangle create with the third side?

It is parallel and half the length of the third side. Signup and view all the answers

What effect does a negative value of 'a' have on the cosine function?

Reflection about the x-axis Signup and view all the answers

Which property is exclusive to squares among quadrilaterals?

Diagonals bisect both pairs of opposite angles. Signup and view all the answers

Which of the following describes the sum of the interior angles of a triangle?

180° Signup and view all the answers

When classifying quadrilaterals, which of the following is a characteristic feature of a trapezium?

One pair of opposite sides is parallel. Signup and view all the answers

What is the range of the function $y = a an heta + q$?

{ f(θ): f(θ) in R } Signup and view all the answers

What is the sum of the interior angles of any quadrilateral?

360° Signup and view all the answers

If two triangles are congruent, which statement must be true?

Their corresponding sides and angles are equal Signup and view all the answers

Which of the following properties does not apply to all rectangles?

Diagonals bisect each other at 90°. Signup and view all the answers

If two adjacent sides of a kite are equal, how is the angle between them characterized?

It can be either acute or obtuse. Signup and view all the answers

Which statement accurately describes an asymptote of the function $y = a an heta + q$?

The asymptotes occur at θ = 90° and θ = 270° Signup and view all the answers

What is the correct relationship between the periods of sine and cosine functions?

Both have a period of 360° Signup and view all the answers

Under which condition can triangles be classified as similar using the SSS criterion?

The corresponding sides are in proportion Signup and view all the answers

How is the y-intercept of the tangent function $y = a an heta + q$ calculated?

y = q Signup and view all the answers

Which triangle classification describes a triangle that has exactly one angle greater than 90°?

Obtuse Signup and view all the answers

In the function $y = a an heta + q$, what does the parameter 'a' influence?

It changes the steepness of the graph branches Signup and view all the answers

What does the line segment connecting the midpoints of two sides of a triangle demonstrate in relation to the third side?

It is parallel to the third side. Signup and view all the answers

If the length of the third side of a triangle is 10 units, what is the expected length of the segment connecting the midpoints of the other two sides?

5 units Signup and view all the answers

In the context of proving geometric properties, how is the mid-point theorem often utilized?

To verify segment proportions and parallelism. Signup and view all the answers

Which statement correctly describes the similarity aspect applied using the mid-point theorem?

It assists in proving similarity in triangles and other geometric shapes. Signup and view all the answers

What role do the angle bisectors play in proving that the quadrilateral MNOP is a parallelogram?

They establish that opposite angles are equal. Signup and view all the answers

Which of the following points correctly represents the gradient of a line if the coordinates of two points are A(-1, 2) and B(3, 6)?

2 Signup and view all the answers

Given the coordinates A(1, 1) and B(3, 3), what is the distance between these two points using the distance formula?

$ ext{2}\sqrt{2}$ Signup and view all the answers

Which property automatically holds true for the sides of a triangle when applying the Mid-Point Theorem?

The sum of lengths of any two sides must be greater than the third. Signup and view all the answers

When plotting a quadrilateral on the Cartesian plane, which order of letters is typically conventional?

Letters should be in alphabetical order. Signup and view all the answers

What is the product of the gradients of two lines that are perpendicular to each other?

-1 Signup and view all the answers

If two points A(2, 3) and B(2, 5) are given, what type of line do these points define?

Vertical line Signup and view all the answers

What is the value of the gradient for a horizontal line?

0 Signup and view all the answers

Which of the following formulas expresses the relationship used to find the equation of a straight line?

y - y₁ = m(x - x₁) Signup and view all the answers

If the midpoint of a line segment between points A(1, 2) and B(3, 4) is calculated, what are the coordinates of the midpoint?

(2, 3) Signup and view all the answers

Which of the following describes parallel lines?

They are always equidistant and have equal gradients. Signup and view all the answers

What is the general form of the equation of a straight line?

y = mx + b Signup and view all the answers

The gradient of which of the following lines is undefined?

A line that runs parallel to the y-axis Signup and view all the answers

Which of the following is the correct trigonometric ratio for cosecant?

Cosec θ = r / y Signup and view all the answers

In which quadrant is the tangent function (tan θ) positive?

Quadrant I and Quadrant III Signup and view all the answers

What is the y-intercept of the function y = 2cosθ?

(0°, 2) Signup and view all the answers

If the sine function is inverted (y = -sin θ), what effect does it have on the graph?

It reflects the graph about the x-axis. Signup and view all the answers

Which statement about the angle of elevation and depression is true?

They are equal when measured from the same point. Signup and view all the answers

What is the sine value for the angle θ where θ = 30°?

$\frac{1}{2}$ Signup and view all the answers

What happens to the range of the function y = a sin θ + q if a = 0?

Range becomes [q, q] Signup and view all the answers

For which special angle is cos θ = 0?

90° Signup and view all the answers

How does the amplitude affect the function y = -3 cos θ?

The graph is stretched vertically by a factor of 3. Signup and view all the answers

What is the period of the function y = cos θ?

360° Signup and view all the answers

What is the length of the hypotenuse in a 45°-45°-90° triangle?

√2 Signup and view all the answers

In a 30°-60°-90° triangle, what is the length of the side opposite the 60° angle?

√3 Signup and view all the answers

Given that θ = 30°, what is the value of sin θ?

1/2 Signup and view all the answers

What angle corresponds to a tangent ratio of √3 in a right triangle?

60° Signup and view all the answers

When finding an unknown angle using the tangent ratio, what is the first step?

Identify the trigonometric ratio involved Signup and view all the answers

If an angle θ measures 123.7°, what is the corresponding angle α if ∠B + α = 180°?

56.3° Signup and view all the answers

Which statement is true regarding the sine and cosine ratios for given angles?

sin 60° is equal to cos 30° Signup and view all the answers

Which trigonometric function would be used to find the angle given the ratio of opposite to adjacent sides?

Tangent Signup and view all the answers

What is the value of cos 60°?

1/2 Signup and view all the answers

What is a condition that indicates there may be no solution to a trigonometric equation?

If values are outside the defined range of the function Signup and view all the answers

In which quadrant would the sine and cosecant functions be negative?

Quadrant III Signup and view all the answers

What is the y-intercept of the function described by $y = 5 an heta + 2$?

2 Signup and view all the answers

Which of the following angles has a sine value equal to $rac{ ext{sqrt}(3)}{2}$?

60° Signup and view all the answers

What is the range of the sine function for $y = -3 ext{sin} heta + 1$?

[-2, 4] Signup and view all the answers

Which of the following correctly represents the tangent function’s value for an angle of 45°?

1 Signup and view all the answers

For the cosine function $y = a ext{cos} heta + q$, if $a < 0$ and $q = 3$, what are the maximum and minimum turning points?

(0, 3), (180, 0) Signup and view all the answers

How does the vertical shift of a sine function affect its graph?

It shifts the graph up or down. Signup and view all the answers

What is the result of $ an 90^ ext{o}$?

undefined Signup and view all the answers

In the context of the CAST diagram, which is true for Quadrant II?

Both sine and cosecant are positive. Signup and view all the answers

What is true about the angles of elevation and depression?

They are equal from corresponding points. Signup and view all the answers

In a 45°-45°-90° triangle, what is the length of the hypotenuse if each leg measures 2 units?

2√2 Signup and view all the answers

What is the angle corresponding to a tangent ratio of $rac{3}{2}$ in the first quadrant?

56.3° Signup and view all the answers

Which of the following is a correct expression for cosine at 60°?

1/2 Signup and view all the answers

If $ an B = rac{3}{2}$ and B is in the second quadrant, what is the approximate value of angle B?

123.7° Signup and view all the answers

From the special angle triangles, what is the ratio of the opposite to the hypotenuse for a 30° angle?

1/2 Signup and view all the answers

Using inverse trigonometric functions, what is the range of possible output angles for $ an^{-1}(x)$?

(-90°, 90°) Signup and view all the answers

What is the sine value for a 45° angle in terms of its special angle triangles?

√2/2 Signup and view all the answers

In a 30°-60°-90° triangle, what is the length of the side opposite the 60° angle if the hypotenuse is measured at 4 units?

2√3 Signup and view all the answers

How is the cosine of a negative angle, specifically cos(-θ), typically represented?

It is equal to cos θ. Signup and view all the answers

If a right-angled triangle has its adjacent side measuring 8 units and the hypotenuse measuring 10 units, what is the cosine of the angle?

0.8 Signup and view all the answers

Given the trigonometric identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$, how can you express cotangent in terms of sine and cosine?

cot \theta = \frac{\cos \theta}{\sin \theta} Signup and view all the answers

Which of the following defines the relationship expressed by the identity $\sin \theta \times \csc \theta = 1$?

The relationship between sine and cosecant Signup and view all the answers

What is the significance of the mnemonic 'Soh Cah Toa' in trigonometry?

It serves as a memory aid for trigonometric ratios Signup and view all the answers

In a right-angled triangle where $ heta$ measures 30 degrees, what is the value of $\tan \theta$?

1/√3 Signup and view all the answers

If $\sec \theta$ is defined as $\frac{1}{\cos \theta}$, what would be the secant of an angle whose cosine is 0.6?

1.667 Signup and view all the answers

For a right-angled triangle with sides measuring 5, 12, and 13, which trigonometric ratio would accurately represent $\cot \theta$ if the opposite side to angle $\theta$ is 5?

2.5 Signup and view all the answers

If the angle $\theta$ is 90 degrees, what can be determined about $\sin \theta$, $\cos \theta$, and $\tan \theta$?

sin = 1, cos = 0, tan is undefined Signup and view all the answers

What is true about the ratios of corresponding sides in similar triangles?

The ratios remain consistent for triangles with the same angles. Signup and view all the answers

Which pair of angle correspondences correctly represents similar triangles?

If \angle A = 30° and \angle D = 30°, then \angle C = 90° and \angle F = 60°. Signup and view all the answers

What is indicated by the notation \Delta ABC \sim \Delta DEF?

Triangle DEF has equal angles to triangle ABC. Signup and view all the answers

In the context of similar triangles, which statement related to corresponding sides is incorrect?

The ratio between corresponding sides is always in descending order. Signup and view all the answers

Which of the following statements accurately describes a property of similar triangles?

Similar triangles have corresponding sides in direct proportion. Signup and view all the answers

If two triangles share a common angle, under what condition will they remain similar?

The other angles must also be equal. Signup and view all the answers

Which ratio exemplifies the relationship between corresponding sides in the mentioned similar triangles?

( \frac{BC}{AC} = \frac{EF}{DF} \) reflects the proportionality in similar triangles. Signup and view all the answers

What is the significance of maintaining the correct order of vertices when labeling similar triangles?

It ensures accurate angle correspondence. Signup and view all the answers

What is the range of the function $y = a an heta + q$?

All real numbers Signup and view all the answers

Which angle corresponds to the location of an asymptote in the tangent function $y = an heta$?

90° Signup and view all the answers

How does a negative value of $a$ affect the graph of the cosine function?

It reflects the graph about the x-axis. Signup and view all the answers

What is the gradient of a line that is vertical?

Undefined Signup and view all the answers

What is the relationship between the gradients of two lines that are perpendicular?

The product of the gradients equals -1. Signup and view all the answers

Which classification describes a triangle with two equal sides and one angle greater than 90°?

Obtuse isosceles Signup and view all the answers

What is the primary condition for two triangles to be considered similar?

All pairs of corresponding angles equal Signup and view all the answers

If the coordinates of two points are A(3, 4) and B(7, 10), what is the gradient between these points?

2 Signup and view all the answers

Which equation indicates a straight line with a gradient of zero?

y = 5 Signup and view all the answers

What happens to the y-intercept of the function $y = a an heta + q$ when $q < 0$?

It shifts down by |q| units. Signup and view all the answers

When comparing the sine and cosine functions, how can the sine function be aligned with the cosine function?

By shifting it 90° to the left Signup and view all the answers

What is the midpoint of a line segment with endpoints at A(4, 6) and B(8, 10)?

(6, 8) Signup and view all the answers

Which triangle congruency rule applies when two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle?

SAS Signup and view all the answers

According to the gradient formula, what happens when the y-coordinates of two points are equal?

The gradient equals zero. Signup and view all the answers

What is the standard form of a straight line represented by the equation y = 3x - 2?

3x - y = 2 Signup and view all the answers

What is the sum of the interior angles in any triangle?

180° Signup and view all the answers

If line WX has a gradient of -3, what is the potential gradient of a line parallel to it?

-3 Signup and view all the answers

What is true about the diagonals of a rhombus?

The diagonals bisect the interior angles and intersect at 90°. Signup and view all the answers

What is the relationship between a square and a rectangle?

A square is a type of rectangle with equal side lengths. Signup and view all the answers

Which of the following best defines a trapezium?

A quadrilateral with at least one pair of opposite sides parallel. Signup and view all the answers

What can be concluded about the angles of a quadrilateral?

The sum of the angles is always 360°. Signup and view all the answers

According to the Mid-Point Theorem, what can be inferred about the line segment joining the mid-points of two sides of a triangle?

It is parallel to the third side and half its length. Signup and view all the answers

What defines a kite in geometry?

A quadrilateral with two pairs of adjacent sides equal. Signup and view all the answers

Which statement about the properties of a parallelogram is incorrect?

All sides are equal in length. Signup and view all the answers

What is a unique property of a rectangle compared to a parallelogram?

All angles are right angles. Signup and view all the answers

In which of the following would one find both pairs of opposite sides equal?

A square. Signup and view all the answers

Which condition must be met for a quadrilateral to be classified as a parallelogram?

Both pairs of opposite sides must be parallel. Signup and view all the answers

What relationship does the line segment joining the mid-points of two sides of a triangle have with the third side?

It is parallel to the third side. Signup and view all the answers

According to the Mid-Point Theorem, how does the length of the line segment joining the mid-points compare to the length of the third side?

It is equal to half the length of the third side. Signup and view all the answers

Which of the following best describes an application of the Mid-Point Theorem?

To prove parallelism in geometric figures. Signup and view all the answers

In a geometric proof involving parallelogram ABCD, which pair of sides is necessarily equal when proving that a certain quadrilateral is a parallelogram?

AB and CD Signup and view all the answers

What is the significance of angle bisectors in parallelogram ABCD?

They establish that opposite angles are equal. Signup and view all the answers

When finding the gradient between two points A(x₁, y₁) and B(x₂, y₂), what does the formula specifically compute?

The vertical component divided by the horizontal component. Signup and view all the answers

Which statement correctly describes the distance formula for points A(x₁, y₁) and B(x₂, y₂)?

Distance is calculated using the square root of the sum of the squares of the differences. Signup and view all the answers

What conclusion can be drawn if two triangles share an angle and their corresponding sides are equal as stated in the proof involving MA and OC?

The triangles are similar by AA. Signup and view all the answers

What does a gradient of 0 indicate about a line on a Cartesian plane?

The line is horizontal. Signup and view all the answers

What is the angle of elevation with respect to an object if the angle of depression from the corresponding point is 40°?

40° Signup and view all the answers

In Quadrant II, which of the following trigonometric ratios is NOT positive?

tan θ Signup and view all the answers

Which of the following statements accurately describes the effect of applying a negative amplitude in the function y = a sin θ + q?

Reflection about the x-axis Signup and view all the answers

What is the value of tan θ at 90°?

undefined Signup and view all the answers

What is the maximum value of y when the function is defined as y = 2 sin θ + 3?

5 Signup and view all the answers

In the function y = cos θ, what are the x-intercepts?

(90°, 0) and (270°, 0) Signup and view all the answers

What is the period of the function y = a cos θ + q?

360° Signup and view all the answers

For a right-angled triangle, which ratio describes the sine function?

The ratio of the opposite side to the hypotenuse Signup and view all the answers

When considering angles in the CAST diagram, which ratio remains positive in Quadrant I?

all ratios Signup and view all the answers

If a triangle has an angle $ heta$ measuring 30° and an opposite side of length 5, what is the length of the hypotenuse?

$10$ Signup and view all the answers

For the tangent function in the form of $y = a \tan \theta + q$, what happens to the graph if $q > 0$?

Shift up by $q$ units. Signup and view all the answers

What is the range of the function $y = a \cos \theta + q$ when $a < 0$?

[q - |a|, q + |a|] Signup and view all the answers

If two triangles are classified as congruent, which statement is necessarily true?

Their corresponding sides are equal. Signup and view all the answers

How can the cosine graph be shifted to align with the sine graph?

Shifted right by 90°. Signup and view all the answers

In a right-angled triangle, if one angle measures 90° and another angle measures 50°, what is the relationship of the third angle?

It must be 40°. Signup and view all the answers

In the context of functions of the form $y = a \tan \theta + q$, what characterizes the steepness of the branches?

The value of $a$. Signup and view all the answers

What is true about the sides of similar triangles?

The sides maintain a constant ratio regardless of the triangles' sizes. Signup and view all the answers

What distinguishes an isosceles triangle from other classifications?

At least two sides are equal in length. Signup and view all the answers

When comparing triangles that are similar, how are their corresponding angles related?

Corresponding angles are equal. Signup and view all the answers

Which criterion allows for proving that two triangles are similar?

Angle-Angle-Angle (AAA) Signup and view all the answers

If two triangles are labeled such that $\Delta ABC \sim \Delta DEF$, what can be concluded about the corresponding sides?

The ratio of the lengths of the sides is equal for all pairs. Signup and view all the answers

What is the effect of the amplitude change when $|a| < 1$ in a cosine function?

Vertical compression. Signup and view all the answers

In a triangle with angles of 30°, 60°, and 90°, what is the constant ratio of the sides opposite these angles?

1:√3:2 Signup and view all the answers

What does the notation $\Delta ABC \sim \Delta DEF$ imply specifically about the triangles?

The order of vertices indicates corresponding sides and angles. Signup and view all the answers

Which of the following ratios must hold true for all sets of similar triangles with corresponding sides?

$\frac{AB}{DE} = \frac{AC}{DF}$ Signup and view all the answers

What fundamental concept forms the basis for many trigonometric concepts?

The similarity of triangles. Signup and view all the answers

If the lengths of corresponding sides of two similar triangles are in the ratio of 4:5, what can be said about their areas?

The areas are in the ratio of 16:25. Signup and view all the answers

What can be concluded about the line segment joining the mid-points of two sides of a triangle?

It is equal in length to half of the third side. Signup and view all the answers

When applying the Mid-Point Theorem, which property is NOT true for the segment joining the mid-points of a triangle's sides?

It divides the triangle into two smaller triangles of equal area. Signup and view all the answers

Which geometric concepts heavily utilize the Mid-Point Theorem in their proofs?

Similarity and segment proportions. Signup and view all the answers

In coordinate geometry, how is the mid-point of a line segment calculated?

By averaging the x-coordinates and y-coordinates of the endpoints. Signup and view all the answers

Which property is true for the line segment that joins the mid-points of two sides of a triangle?

It can be used to verify properties of polygon angles. Signup and view all the answers

What is the primary use of the Mid-Point Theorem in proof construction?

To establish relationships between different triangles. Signup and view all the answers

Given a triangle with vertices A, B, and C, how is the line segment joining the mid-points D and E, of sides AB and AC respectively, characterized?

It is parallel to side BC. Signup and view all the answers

In the context of similarity in triangles, what role does the line joining mid-points of two sides serve?

It substantiates claims of proportionality in segments. Signup and view all the answers

What is the relationship of the length of the segment that joins the mid-points of two sides of a triangle to the length of the third side?

It is exactly half the length of the third side. Signup and view all the answers

Which application does the Mid-Point Theorem NOT support in geometry?

Calculating the area of complex quadrilaterals. Signup and view all the answers

In a 30°-60°-90° triangle, what is the length of the hypotenuse if the length of the side opposite the 30° angle is 2?

4 Signup and view all the answers

What is the value of tan 30°?

√3/3 Signup and view all the answers

How do you express sin 45° in simplest radical form?

√2/2 Signup and view all the answers

For the angle B found from Q(-2, 3) using the tangent ratio, which of the following correctly represents the calculation?

tan B = 3/2 Signup and view all the answers

What is the angle formed by the line from the origin to point P(2, 3) with the positive x-axis?

56.3° Signup and view all the answers

Which limit statement accurately describes the domain of the sine function?

|sin θ| ≤ 1 Signup and view all the answers

In solving for an unknown angle using trigonometric equations, which inverse function would be used to find the angle if sin θ = 1/2?

sin^-1(1/2) Signup and view all the answers

For a 45°-45°-90° triangle, what is the relationship between the lengths of the sides and their hypotenuse?

Both legs are equal and the hypotenuse is √2 times the leg length. Signup and view all the answers

What is the approximate value of cos 60°?

1/2 Signup and view all the answers

If an angle θ corresponds to a ratio of tan θ = √3, which of the following angles could θ be?

60° Signup and view all the answers

What is the value of $\tan \theta$ if the length of the side opposite $\theta$ is 6 and the length of the adjacent side is 8?

$\frac{6}{8}$ Signup and view all the answers

If $\sin \theta = 0.5$, what is the value of $\text{cosec } \theta$?

2 Signup and view all the answers

Which of the following statements about $\cos \theta$ is true?

$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ Signup and view all the answers

Which of the following expressions represents the reciprocal of $\tan \theta$?

$\cot \theta$ Signup and view all the answers

In a right triangle where $\sin 60^{\circ} = \frac{\sqrt{3}}{2}$, what is the value of $\text{cosec } 60^{\circ}$?

2 Signup and view all the answers

Which of the following trigonometric ratios is equal to 1?

$\cos \theta \times \sec \theta$ Signup and view all the answers

If a right triangle has an adjacent side of length 5 and a hypotenuse of length 13, what is the value of $\cos \theta$?

$\frac{5}{13}$ Signup and view all the answers

What is the value of $\tan 45^{\circ}$?

1 Signup and view all the answers

How does the Pythagorean theorem relate to trigonometric ratios in a right triangle?

It confirms that $\sin^2 \theta + \cos^2 \theta = 1$. Signup and view all the answers

In which scenario can a calculator directly compute the value of $\sec \theta$?

When you enter $\frac{1}{\cos \theta}$ directly. Signup and view all the answers

What is true about the diagonals of a rhombus?

The diagonals intersect at 90°. Signup and view all the answers

Which property differentiates a rectangle from a square?

All sides are equal in length. Signup and view all the answers

What can be inferred about the angles in a trapezium?

Interior angles sum to 360°. Signup and view all the answers

What is a unique characteristic of a kite among the different types of quadrilaterals?

It has two pairs of adjacent sides equal. Signup and view all the answers

Which statement describes the relationship between a rhombus and a parallelogram?

Every rhombus is a parallelogram. Signup and view all the answers

In the context of the Mid-Point Theorem, what can be concluded about the segment joining the mid-points of two sides of a triangle?

It is parallel to the third side. Signup and view all the answers

Which of the following statements correctly describes a square?

It is a rhombus with all angles equal to 90°. Signup and view all the answers

How does the sum of the interior angles of a quadrilateral compare to that of a triangle?

It is greater than that of a triangle by 180°. Signup and view all the answers

In what way does a trapezium differ from a parallelogram?

A trapezium can have non-parallel adjacent sides. Signup and view all the answers

What is the gradient of a line defined by points A(1, 2) and B(4, 6)?

$\frac{4}{3}$ Signup and view all the answers

What is the equation of a line in slope-intercept form with a gradient of -2 and a y-intercept of 5?

$y = -2x + 5$ Signup and view all the answers

Which relationship describes two lines that are perpendicular to each other?

The product of their gradients is equal to -1. Signup and view all the answers

What is the mid-point of a line segment connecting points C(3, 7) and D(5, 1)?

(4, 4) Signup and view all the answers

If the gradient of a horizontal line is zero, what can be said about its y-coordinate?

It can be any real number. Signup and view all the answers

Given two points, P(2, 5) and Q(2, 3), what type of line do they form?

A vertical line Signup and view all the answers

What is the correct formula to find the gradient of line segment AB given points A(3, 4) and B(3, 8)?

Undefined Signup and view all the answers

If two lines have gradients of 1/2 and -2, what relationship do these lines have?

They are perpendicular. Signup and view all the answers

Gr 10 Math June P2 Hard

Questions and Answers

What can be inferred about the lengths of corresponding sides in similar triangles?

Which of the following statements about the angles in similar triangles is true?

How does the order of the vertices affect the similarity of two triangles?

What is a key characteristic of the ratios of corresponding sides in similar triangles?

In the case of triangles $ riangle ABC$, $ riangle DEF$, and $ riangle GHK$, which ratio is expected to be consistent?

Which statement correctly describes the concept of trigonometric ratios?

What is the fundamental requirement for triangles to be considered similar?

Which of the following ratios illustrates the relationship between similar triangles accurately?

What effect does an amplitude change with property |a| > 1 have on the cosine function?

Which classification of triangle describes a triangle with all sides of different lengths?

When comparing the sine and cosine functions, how can the cosine graph be visually aligned with the sine graph?

What is the range of the function y = a cos θ + q when a > 0?

For the tangent function y = a tan θ + q, what does the variable q represent?

Which statement about a triangle's exterior angles is correct?

Which rule would be used to prove two triangles congruent if two sides and the included angle are equal?

In the context of triangle similarity, what does the SSS criterion specifically require?

Which of the following describes the y-intercept of the function y = a tan θ + q?

What is the period of the tangent function y = tan θ?

Which of the following statements is not a property of a parallelogram?

What can be concluded if a quadrilateral is identified as a kite?

What is the relationship between a square and a rectangle?

What is the value of the sine function at an angle of 45°?

In a 30°-60°-90° triangle, what is the ratio of the length of the hypotenuse to the length of the side opposite the 30° angle?

According to the Mid-Point Theorem, what is true about the segment connecting the mid-points of two sides of a triangle?

What is the tangent of angle θ if the opposite side is 4 and the adjacent side is 2?

Which property is unique to a rhombus among quadrilaterals?

If a triangle has sides measuring 3, 4, and 5, what can be deduced about this triangle?

Which of the following angles has a cosine value of 1/2?

When using the inverse sine function, what is the valid range of outputs?

What characterizes a trapezium in relation to its sides?

If $\tan B = \frac{3}{2}$ for point Q(-2, 3), what is the value of angle B?

How do the angles of a quadrilateral compare to each other?

Which statement is true regarding the diagonals of a rectangle?

What is the relationship between the sides of a 45°-45°-90° triangle?

What restrictions apply to the values of sine and cosine functions?

In the context of trigonometric ratios, what does the tangent function represent?

What is the gradient of a line that runs parallel to the x-axis?

What can be said about the gradients of two perpendicular lines?

To find the midpoint of a line segment between points A(4, 2) and B(6, 8), what are the coordinates of the midpoint?

What is the slope formula used for determining the gradient of a line between two points?

If line WX is perpendicular to line YZ and the gradient of line WX is 2, what is the gradient of line YZ?

Given points A(2, 3) and B(2, 5), what type of line do these points define?

Which of the following statements about gradients is true?

Which equation represents the standard form of a straight line?

What is the value of $\tan 60^\circ$?

In which quadrant is $\sin \theta$ positive and $\tan \theta$ negative?

If $a = -2$ and $q = 3$, what is the range of the function $y = a \sin \theta + q$?

What is the period of the function $y = \cos \theta$?

Which of the following values corresponds to $\cos 30^\circ$?

What is the y-intercept of the function $y = -3 \sin \theta + 2$?

Which trigonometric ratio is defined as $\frac{\text{adjacent}}{\text{hypotenuse}}$?

What is the correct definition of the tangent ratio in a right-angled triangle?

Under which condition is the value of $\tan \theta$ undefined?

Which of the following statements about the trigonometric ratios is true?

What is the reciprocal of the cosine ratio?

What is the angle of elevation equal to when viewed from the same horizontal plane down to the ground?

When converting $ ext{cosec } heta$ to a standard trigonometric ratio, which expression is used?

Which pair of trigonometric ratios is reciprocally related?

For angle $ heta$, what does the relationship $ ext{sin } heta imes ext{cosec } heta = 1$ imply?

If a calculator is used to find $ ext{sec } 34^ ext{o}$, which standard trigonometric ratio should be used?

Which angle does not require a calculator to determine its trigonometric ratio?

Which relation holds true among the trigonometric identities?

When employing the mnemonic 'Soh Cah Toa', what concept does 'Cah' represent?

What does the line segment joining the mid-points of two sides of a triangle represent in relation to the third side?

Which property is NOT used to prove that MNOP is a parallelogram?

In the proof for parallelogram MNOP, which pair of triangles are shown to be congruent using the AAS criterion?

How is the distance formula derived for calculating the distance between two points in a Cartesian plane?

What characterizes the gradient of a line between two points?

Which feature of a quadrilateral must be proven to classify it as a parallelogram?

What is the necessary conclusion about the angles in parallelogram MNOP?

To find the gradient of the line between two points A(x₁, y₁) and B(x₂, y₂), which values are used?

Which of the following would NOT be a valid application of the Mid-Point Theorem?

When plotting a quadrilateral ABCD on the Cartesian plane, what is crucial about the order of vertices?

When comparing the ratios of corresponding sides of two similar triangles, what will be true regardless of the triangles' sizes?

Which of the following statements accurately describes the importance of corresponding angles in similar triangles?

If triangles $Δ ABC$ and $Δ DEF$ are similar with a given proportion of the sides, how would you express the relationship of the sides mathematically?

What must be true about the order of vertices when determining triangle similarity?

What is the significance of calculating corresponding side ratios in similar triangles?

If the sine of an angle is $rac{3}{5}$, what is the value of cosecant at that angle?

If the cosecant of an angle is $rac{7}{2}$, what is the sine of that angle?

From the trigonometric ratios, if the adjacent side measures 5 and the hypotenuse measures 13, what is $rac{ ext{adjacent}}{ ext{hypotenuse}}$?