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Questions and Answers
What is the value of OH according to Pythagoras' Theorem when sin 20° = OH / 5?
What is the value of OH according to Pythagoras' Theorem when sin 20° = OH / 5?
Which theorem relates to the relationship of angles and sides in the given geometric configuration?
Which theorem relates to the relationship of angles and sides in the given geometric configuration?
If  = 2x and B̂3 = 2x, what can be concluded about angles  and B̂3?
If  = 2x and B̂3 = 2x, what can be concluded about angles  and B̂3?
What relationship does CV have with AT in the isosceles trapezium BTVC?
What relationship does CV have with AT in the isosceles trapezium BTVC?
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What can be inferred when T̂1 = V̂1 + V̂2?
What can be inferred when T̂1 = V̂1 + V̂2?
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If angles  and B̂3 both equal 2x, which property do they exhibit?
If angles  and B̂3 both equal 2x, which property do they exhibit?
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What is the significance of angles being opposite equal sides in a triangle?
What is the significance of angles being opposite equal sides in a triangle?
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Which statement best describes the configuration of BTVC?
Which statement best describes the configuration of BTVC?
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What is the length of the CD calculated using the formula $CD = \frac{121 \times \sin(9.2^\circ)}{\sin(135^\circ)}$?
What is the length of the CD calculated using the formula $CD = \frac{121 \times \sin(9.2^\circ)}{\sin(135^\circ)}$?
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If the volume of the shape is represented by the equation $V = x^2h$ and $V = 490$, what is the expression for $h$ in terms of $x$?
If the volume of the shape is represented by the equation $V = x^2h$ and $V = 490$, what is the expression for $h$ in terms of $x$?
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What is the surface area calculated when $x = 7$?
What is the surface area calculated when $x = 7$?
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What is the derived formula for the base area including both $x^2$ and $4xh$?
What is the derived formula for the base area including both $x^2$ and $4xh$?
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Substituting $h = 10$ into the volume equation $10 = \frac{490}{x^2}$ results in what equation?
Substituting $h = 10$ into the volume equation $10 = \frac{490}{x^2}$ results in what equation?
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If the equation for area includes $4x \left( \frac{490}{x} \right)$, what does this simplify to?
If the equation for area includes $4x \left( \frac{490}{x} \right)$, what does this simplify to?
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Given the dimensions and formulas, how is $h$ maximized when expressed in terms of $x$?
Given the dimensions and formulas, how is $h$ maximized when expressed in terms of $x$?
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What is the significance of the equation $490 = x^2h$ in relation to the shape's volume?
What is the significance of the equation $490 = x^2h$ in relation to the shape's volume?
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What is the range of x that satisfies the inequality $-90^{ ext{o}} < x < -45^{ ext{o}}$?
What is the range of x that satisfies the inequality $-90^{ ext{o}} < x < -45^{ ext{o}}$?
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Which of the following is equivalent to $h(x) = \frac{\sin x}{\cos x} + 1$?
Which of the following is equivalent to $h(x) = \frac{\sin x}{\cos x} + 1$?
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What is the maximum value of the function $p(x) = -2 \cos(2x)$?
What is the maximum value of the function $p(x) = -2 \cos(2x)$?
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For the angle $p$ in the equation $\tan p = \frac{3}{3}$, what is the value of $p$ in degrees rounded to three decimal places?
For the angle $p$ in the equation $\tan p = \frac{3}{3}$, what is the value of $p$ in degrees rounded to three decimal places?
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Which equation represents a vertical translation of the function $h(x) = \tan x$?
Which equation represents a vertical translation of the function $h(x) = \tan x$?
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In triangle BCD, if $CD = 121$ and angle $p = 18.4^{ ext{o}}$, which relationship is valid for the sine rule?
In triangle BCD, if $CD = 121$ and angle $p = 18.4^{ ext{o}}$, which relationship is valid for the sine rule?
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If $f(x) = g(x)$ at $x = 45^{ ext{o}}$, what can be deduced about the functions at that point?
If $f(x) = g(x)$ at $x = 45^{ ext{o}}$, what can be deduced about the functions at that point?
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What transformation does the equation $h(x) = \tan x + 2$ indicate?
What transformation does the equation $h(x) = \tan x + 2$ indicate?
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What is the value of k when solving the equation $7k - 2 = 75$?
What is the value of k when solving the equation $7k - 2 = 75$?
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What is the sum of the coefficients in the expression $2k + k + 1 + k + 2 + k - 3 + 2k - 2$?
What is the sum of the coefficients in the expression $2k + k + 1 + k + 2 + k - 3 + 2k - 2$?
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If the mean of a set of data points decreases by t units, which statement is correct regarding the standard deviation?
If the mean of a set of data points decreases by t units, which statement is correct regarding the standard deviation?
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When manipulating the equation $\frac{7k - 2}{5} = 15$, what is the first step to isolate the term with k?
When manipulating the equation $\frac{7k - 2}{5} = 15$, what is the first step to isolate the term with k?
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Which of the following represents the standard deviation of the data points 22, 12, 13, 8, 20?
Which of the following represents the standard deviation of the data points 22, 12, 13, 8, 20?
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What is the measure of angle $x$ in triangle ATB if $5x = 180°$?
What is the measure of angle $x$ in triangle ATB if $5x = 180°$?
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In the context of a cyclic quadrilateral, which relationship holds true for exterior angle D^3?
In the context of a cyclic quadrilateral, which relationship holds true for exterior angle D^3?
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What can be concluded if H^1 = F^2 and H^1 = F^1?
What can be concluded if H^1 = F^2 and H^1 = F^1?
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According to the tan chord theorem, if H^1 = F^2, which angle must also be equal?
According to the tan chord theorem, if H^1 = F^2, which angle must also be equal?
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If K^1 = E^1 and the exterior angle is related to angles in the same circle segment, what can be derived for K^2?
If K^1 = E^1 and the exterior angle is related to angles in the same circle segment, what can be derived for K^2?
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What is true regarding angles H^3 and H^4 in relation to angles E^1 and E^2 in a cyclic quadrilateral?
What is true regarding angles H^3 and H^4 in relation to angles E^1 and E^2 in a cyclic quadrilateral?
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Which theorem is being applied when stating that F^1 and F^2 are equal based on angles in the same segment?
Which theorem is being applied when stating that F^1 and F^2 are equal based on angles in the same segment?
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In triangle ATB, which representation of angle D^3 highlights its connection to angles in cyclic quadrilaterals?
In triangle ATB, which representation of angle D^3 highlights its connection to angles in cyclic quadrilaterals?
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Study Notes
Question 7
- The tangent of an angle (p) is 1/3.
- The value of p is 18.435 degrees, calculated to 3 decimal places.
- In triangle BCD, the sine rule can be used to find the length of CD using the known values of angle p, BC, and angle B.
- CD = 27.36 meters.
Question 8
- The volume of a rectangular prism is 490 cubic centimeters.
- The height (h) of the prism can be expressed as 490/x^2, where x is the side length of the square base.
- The surface area of the prism is calculated by adding the area of the square base (x^2) and the area of the four rectangular sides (4xh).
- The surface area can be simplified to x^2 + 1960/x.
- When the height is 10 cm, the side length of the base (x) is 7 cm.
- The total surface area of the prism is then 329 cm².
Question 9
- In the diagram, point O is the center of the circle and OE is a radius.
- OH is perpendicular to chord RS.
- Using Pythagoras' Theorem, the length of OH is 1.71 units.
- OR is a radius of the circle, so OE and OR are equal to 5 units.
- Using the sine function, sin(20°) = OH/OE, we can find OH = 1.71 units.
Question 10
- Angle B1 is equal to angle V1 because they are alternate angles formed by parallel lines AC and TV.
- Angle V1 is equal to angle T2 because they are angles opposite the equal sides of triangle VBT.
- Angle T2 is equal to angle B1 due to the tangent chord theorem.
- Angle A is double angle B1 because they are opposite angles of a parallelogram.
- Angle B3 is double angle B1 because it is an exterior angle of cyclic quadrilateral ATVB.
- Therefore, angle A is equal to angle B3, both equal 2x.
- This means AT = BT, as sides opposite equal angles in a triangle are equal.
- Alternatively, angle B3 is double angle B1 (exterior angle of cyclic quadrilateral), angle T1 is double angle B1 (alternate angles, AC||TV), and therefore T1 = V1+ V2, both equal 2x.
- This suggests that BTVC is an isosceles trapezium because it has one pair of parallel sides and one pair of equal base angles.
- Therefore, CV = BT and CV = AT, leading to the conclusion that AT = BT.
- In triangle ATB, the sum of angles is 180 degrees.
- Solving for x, we get x = 36 degrees.
Question 11
- Angle D3 is equal to the sum of angles H1 and E1, as it is an exterior angle of triangle DHE.
- Angle H1 is equal to angle F2 due to the tangent chord theorem.
- Angle E1 is equal to angle F1 due to the tangent chord theorem.
- Therefore, angle D3 is equal to the sum of angles F1 and F2.
- This indicates that quadrilateral DEFH is cyclic because its exterior angle (D3) is equal to the sum of the opposite interior angles (F1 and F2).
- Angle H1 is equal to angle F2 and F1 because they are angles in the same segment of the circle.
- Therefore, angle F1 is equal to angle F2.
- In the cyclic quadrilateral DEFH, angle H3 + angle H4 = angle E1 + angle E2.
- Angle H4 is equal to angle K1 using the tangent chord theorem.
- Angle H3 is equal to angle K2 because they are angles in the same segment of the circle.
- Therefore, angle K1 + angle K2 is equal to angle E1 + angle E2.
- Given that angle K1 is equal to angle E1, then angle K2 is equal to angle E2.
- This proves that KF is a tangent to the circle according to the converse of the tangent chord theorem (if angle K2 = angle E2).
Question 2
- To find the value of k, substitute the given expressions for the five terms and simplify the equation.
- Solve for k, obtaining k = 11.
- The dataset with the given values is: 22, 12, 13, 8, and 20.
- The standard deviation of this dataset is 5.22.
- If each value in the dataset is decreased by "t" units, the mean will decrease by "t" units, but the standard deviation will remain unchanged.
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Description
Test your knowledge on geometry and trigonometry concepts, including the tangent of angles, volume and surface area of prisms, and properties of circles. This quiz covers various mathematical principles and problem-solving techniques. Challenge yourself with these thought-provoking questions!
