Gr12 Mathematics: June mix P(2)
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Questions and Answers

What is the equation of a circle with center at the origin and radius r?

  • x^2 + y^2 = r^2 (correct)
  • x^2 - y^2 = r^2
  • x^2 / y^2 = r^2
  • x^2 * y^2 = r^2

    • What is a property of a circle with center at the origin?

  • It is asymmetric about the origin.
  • It is asymmetric about the y-axis.
  • It is asymmetric about the x-axis.
  • It is symmetric about the x-axis and y-axis. (correct)

    • What is the equation of a circle with center at (a, b) and radius r?

  • (x - a)^2 - (y - b)^2 = r^2
  • (x - a)^2 / (y - b)^2 = r^2
  • (x - a)^2 * (y - b)^2 = r^2
  • (x - a)^2 + (y - b)^2 = r^2 (correct)

    • What is the purpose of completing the square in the equation of a circle?

    <p>To rewrite the equation in standard form.</p> Signup and view all the answers

    What is the first step in completing the square to find the center and radius of a circle?

    <p>Group the x terms and the y terms.</p> Signup and view all the answers

    What is the general form of a circle's equation?

    <p>x^2 + y^2 + Dx + Ey + F = 0</p> Signup and view all the answers

    What is the advantage of completing the square?

    <p>It allows us to find the center and radius of the circle.</p> Signup and view all the answers

    What is used to derive the equation of a circle with center at the origin?

    <p>Pythagorean theorem</p> Signup and view all the answers

    What is the formula for the area of a triangle?

    <p>Area = 1/2 × base × height</p> Signup and view all the answers

    What is the Mid-point Theorem?

    <p>The line joining the midpoints of two sides of a triangle is parallel to the third side and equal to half its length.</p> Signup and view all the answers

    What is the condition for two polygons to be similar?

    <p>They have the same number of sides and all pairs of corresponding angles and sides are in the same proportion.</p> Signup and view all the answers

    What is the Proportion Theorem?

    <p>A line drawn parallel to one side of a triangle divides the other two sides proportionally.</p> Signup and view all the answers

    What is the Converse of the Mid-point Theorem?

    <p>The line drawn from the midpoint of one side of a triangle parallel to another side bisects the third side.</p> Signup and view all the answers

    What is the condition for two triangles to be similar?

    <p>They have the same number of sides and all pairs of corresponding angles and sides are in the same proportion.</p> Signup and view all the answers

    What is the formula for the proportion theorem?

    <p>AD/DB = AE/EC</p> Signup and view all the answers

    What is the theorem that states equiangular triangles are similar?

    <p>The Theorem of Equiangular Triangles</p> Signup and view all the answers

    What is the condition for triangles on the same base and equal in area?

    <p>They lie between parallel lines.</p> Signup and view all the answers

    What is the definition of similar polygons?

    <p>Polygons with the same shape but different sizes.</p> Signup and view all the answers

    What is the gradient of the radius of a circle?

    <p>The slope of the line from the center of the circle to the point of tangency.</p> Signup and view all the answers

    Which of the following is NOT a property of proportion?

    <p>Inverse proportion</p> Signup and view all the answers

    What is the standard form of the equation of a circle?

    <p>(x - a)^2 + (y - b)^2 = r^2</p> Signup and view all the answers

    What is the relationship between the gradient of the radius and the gradient of the tangent line at the point of tangency?

    <p>They are negative reciprocals.</p> Signup and view all the answers

    If a line is drawn parallel to one side of a triangle, what does it do to the other two sides?

    <p>It divides the other two sides proportionally.</p> Signup and view all the answers

    What is the equation of a tangent line to a circle given the center of the circle and the point of tangency?

    <p>y - y1 = m(x - x1)</p> Signup and view all the answers

    What is a tangent line to a circle?

    <p>A line that touches the circle at exactly one point without crossing it.</p> Signup and view all the answers

    What is the gradient of the line from the center of a circle to the point of tangency called?

    <p>Slope of the radius</p> Signup and view all the answers

    What is the relationship between the gradient of the radius and the gradient of the tangent line at the point of tangency?

    <p>They are negative reciprocals.</p> Signup and view all the answers

    What is the standard form of the equation of a circle?

    <p>(x - a)^2 + (y - b)^2 = r^2</p> Signup and view all the answers

    What is the formula for the area of a parallelogram?

    <p>$Area = base imes height$</p> Signup and view all the answers

    What is the name of the theorem that states if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally?

    <p>Basic Proportionality Theorem</p> Signup and view all the answers

    What is the relationship between the areas of two triangles with the same height?

    <p>The areas are proportional to their bases.</p> Signup and view all the answers

    What is the formula for the area of a trapezoid?

    <p>$Area = rac{1}{2} imes (base_1 + base_2) imes height$</p> Signup and view all the answers

    Which of the following statements is true about similar polygons?

    <p>Corresponding angles are equal, and corresponding sides are in proportion.</p> Signup and view all the answers

    Which of the following polygons has a formula for its area that involves multiplying its two diagonals?

    <p>Rhombus</p> Signup and view all the answers

    What is the relationship between the areas of two triangles with equal bases and between the same parallel lines?

    <p>The areas are equal.</p> Signup and view all the answers

    Which of the following is NOT a property of a kite?

    <p>All four sides are equal in length.</p> Signup and view all the answers

    Which theorem states that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides?

    <p>Pythagoras' Theorem</p> Signup and view all the answers

    What is the formula for the area of a square?

    <p>$Area = side^2$</p> Signup and view all the answers

    Which of the following is the correct formula for the sine of a sum?

    <p>sin(α + β) = sin α cos β + cos α sin β</p> Signup and view all the answers

    Which of the following is a correct form of the cosine of a double angle formula?

    <p>cos(2α) = 1 - 2sin²α</p> Signup and view all the answers

    What is the first step in deriving the double angle formula for sine?

    <p>Use the sine of a sum formula.</p> Signup and view all the answers

    Which of the following is NOT a valid formula for the cosine of a double angle?

    <p>cos(2α) = 1 + 2sin²α</p> Signup and view all the answers

    What is the purpose of using a CAST diagram when solving trigonometric equations?

    <p>To determine the quadrant where the solution lies.</p> Signup and view all the answers

    Which of the following is the correct formula for the sine of a difference?

    <p>sin(α - β) = sin α cos β - cos α sin β</p> Signup and view all the answers

    What is the formula for the cosine of a difference?

    <p>cos(α - β) = cos α cos β + sin α sin β</p> Signup and view all the answers

    Which of the following is NOT a valid step in the general solution method for trigonometric equations?

    <p>Find the solutions within a specified interval by adding or subtracting multiples of the period.</p> Signup and view all the answers

    What is the double angle formula for cosine in terms of sine?

    <p>cos(2α) = 1 - 2sin²α</p> Signup and view all the answers

    Which of the following is the formula for the cosine of a sum?

    <p>cos(α + β) = cos α cos β - sin α sin β</p> Signup and view all the answers

    Which trigonometric rule is used when no perpendicular height is given in a triangle?

    <p>Area Rule</p> Signup and view all the answers

    When is the Sine Rule used?

    <p>When two angles and a side are given</p> Signup and view all the answers

    What is the formula for the area of triangle ABC?

    <p>All of the above</p> Signup and view all the answers

    Which trigonometric rule is used to find the length of a side in a triangle when two sides and the included angle are given?

    <p>Cosine Rule</p> Signup and view all the answers

    If (\sin \theta = x), which of the following represents the general solution for (\theta)?

    <p>$\theta = \sin^{-1} x + k \cdot 360^\circ$</p> Signup and view all the answers

    Which of the following is the general solution for (\theta) if (\cos \theta = x)?

    <p>$\theta = \cos^{-1} x + k \cdot 360^\circ$</p> Signup and view all the answers

    What is the general solution for (\theta) when (\tan \theta = x)?

    <p>$\theta = an^{-1} x + k \cdot 180^\circ$</p> Signup and view all the answers

    Which of the following is the correct formula for the Cosine Rule?

    <p>$\a^2 = b^2 + c^2 - 2bc \cos A$</p> Signup and view all the answers

    What must be proven for triangles to establish similarity?

    <p>Equiangularity or proportionality of sides</p> Signup and view all the answers

    Which formula represents the cosine of a difference?

    <p>cos(α - β) = cos α cos β + sin α sin β</p> Signup and view all the answers

    Which condition is necessary for polygons to be similar?

    <p>All corresponding angles are equal and all corresponding sides are in the same proportion</p> Signup and view all the answers

    What is the cosine of a sum formula?

    <p>cos(α + β) = cos α cos β - sin α sin β</p> Signup and view all the answers

    What is the correct expression for sin(α - β)?

    <p>sin(α - β) = sin α cos β - cos α sin β</p> Signup and view all the answers

    Which identity is used in the derivation of cos(α + β)?

    <p>Negative angle identity</p> Signup and view all the answers

    How is cos(α - β) derived using the distance formula?

    <p>By using coordinates on the unit circle</p> Signup and view all the answers

    Which of the following best describes sine addition?

    <p>sin(α + β) = sin α cos β + cos α sin β</p> Signup and view all the answers

    Which of the following statements is true about triangles with sides in proportion?

    <p>They are always similar.</p> Signup and view all the answers

    In the proof of the Pythagorean Theorem, why is (\triangle ABD) similar to (\triangle CBA)?

    <p>AAA similarity criterion</p> Signup and view all the answers

    If two triangles have equal bases between the same parallel lines, what can be said about their areas?

    <p>They have equal areas.</p> Signup and view all the answers

    In the proof of the theorem 'Triangles with Sides in Proportion are Similar', why is (GH) parallel to (BC)?

    <p>Sides are in proportion.</p> Signup and view all the answers

    What is the key concept used to prove the similarity of triangles in the text?

    <p>Angle-Angle-Angle (AAA) criterion.</p> Signup and view all the answers

    Which of these is NOT a direct consequence of the proportionality theorem?

    <p>If the corresponding sides of two triangles are in proportion, then the two triangles are similar.</p> Signup and view all the answers

    What is the purpose of constructing line segment (GH) in the proof of the theorem 'Triangles with Sides in Proportion are Similar'?

    <p>To prove that (\triangle AGH) is similar to (\triangle ABC).</p> Signup and view all the answers

    What is the formula for the area of a triangle?

    <p>Area = base × height / 2</p> Signup and view all the answers

    What is the name of the theorem that states if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally?

    <p>Basic Proportionality Theorem</p> Signup and view all the answers

    What is the formula for the area of a parallelogram?

    <p>Area = base × height</p> Signup and view all the answers

    What is the relationship between the areas of two triangles with the same height?

    <p>They are proportional to their bases</p> Signup and view all the answers

    What is the formula for the area of a rhombus?

    <p>Area = diagonal1 × diagonal2 / 2</p> Signup and view all the answers

    What is the formula for the area of a trapezium?

    <p>Area = (base1 + base2) × height / 2</p> Signup and view all the answers

    What is the relationship between the areas of two triangles with equal bases and between the same parallel lines?

    <p>They are equal</p> Signup and view all the answers

    What is the formula for the area of a kite?

    <p>Area = diagonal1 × diagonal2 / 2</p> Signup and view all the answers

    Which of the following polygons has a formula for its area that involves multiplying its two diagonals?

    <p>Kite</p> Signup and view all the answers

    What is the name of the theorem that states that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides?

    <p>Pythagoras' Theorem</p> Signup and view all the answers

    What is the equation of a circle with center at (3, -2) and radius 5?

    <p>(x - 3)^2 + (y + 2)^2 = 25</p> Signup and view all the answers

    Which of the following is NOT a symmetry property of a circle with center at the origin?

    <p>Symmetry about the line y = 2x</p> Signup and view all the answers

    What is the radius of the circle represented by the equation x^2 + y^2 - 6x + 4y + 4 = 0?

    <p>3</p> Signup and view all the answers

    What is the center of the circle represented by the equation x^2 + y^2 + 8x - 10y + 16 = 0?

    <p>(-4, 5)</p> Signup and view all the answers

    Which of the following is the correct step to complete the square for the x terms in the equation x^2 - 10x + y^2 + 6y = 12?

    <p>Add 25 to the left side and 25 to the right side</p> Signup and view all the answers

    What is the equation of the circle with center at the origin and passing through the point (4, 3)?

    <p>x^2 + y^2 = 25</p> Signup and view all the answers

    The equation of a circle is given as (x + 2)^2 + (y - 1)^2 = 9. What is the center of the circle?

    <p>(-2, 1)</p> Signup and view all the answers

    Which of the following points lies on the circle with equation (x - 1)^2 + (y + 3)^2 = 16?

    <p>(5, -3)</p> Signup and view all the answers

    If a line is drawn parallel to one side of a triangle, how does it divide the other two sides?

    <p>It divides the other two sides in the same proportion.</p> Signup and view all the answers

    Two triangles are considered similar if they have:

    <p>The same shape but different sizes.</p> Signup and view all the answers

    What is the relationship between the areas of two triangles with equal bases and between the same parallel lines?

    <p>The areas of the two triangles are equal.</p> Signup and view all the answers

    What is the relationship between the areas of two triangles with equal heights?

    <p>The areas of the two triangles are proportional to their bases.</p> Signup and view all the answers

    If two triangles are equiangular, what is the relationship between their corresponding sides?

    <p>The corresponding sides are proportional.</p> Signup and view all the answers

    The line joining the midpoints of two sides of a triangle is:

    <p>Parallel to the third side and equal to half the third side.</p> Signup and view all the answers

    Which of the following is NOT a condition for two polygons to be similar?

    <p>The polygons have the same shape.</p> Signup and view all the answers

    Two triangles on the same side of the same base and equal in area lie:

    <p>Between two parallel lines.</p> Signup and view all the answers

    If a line drawn from the midpoint of one side of a triangle is parallel to another side, what does it do to the third side?

    <p>It divides the third side into equal lengths.</p> Signup and view all the answers

    What is the relationship between the gradients of the tangent line and the radius of a circle at the point of tangency?

    <p>They are perpendicular.</p> Signup and view all the answers

    What is the compound angle identity for (\cos(\alpha - \beta))?

    <p>(\cos \alpha \cos \beta + \sin \alpha \sin \beta)</p> Signup and view all the answers

    Which of the following is used to rewrite (\cos(\alpha + \beta)) as a difference of angles for deriving its compound angle identity?

    <p>Negative angle identity</p> Signup and view all the answers

    Which compound angle identity can be used to directly derive (\cos(\alpha + \beta))?

    <p>(\cos(\alpha - \beta))</p> Signup and view all the answers

    The derivation of (\cos(\alpha - \beta)) involves equating two expressions. What are these expressions?

    <p>Distance formula and cosine rule</p> Signup and view all the answers

    What is the compound angle identity for (\sin(\alpha - \beta))?

    <p>(\sin \alpha \cos \beta - \cos \alpha \sin \beta)</p> Signup and view all the answers

    Which of the following identities is NOT used in the derivation of (\cos(\alpha + \beta))?

    <p>Double angle identity</p> Signup and view all the answers

    What is the compound angle identity for (\sin(\alpha + \beta))?

    <p>(\sin \alpha \cos \beta + \cos \alpha \sin \beta)</p> Signup and view all the answers

    In the derivation of (\cos(\alpha - \beta)), what is the relationship between the points K and L on the unit circle?

    <p>They are the endpoints of a chord</p> Signup and view all the answers

    What is the formula for the radius of the circle in standard form?

    <p>r = rac{D^2 + E^2}{4} - F</p> Signup and view all the answers

    Which formula correctly relates the gradients of the radius and the tangent line at the point of tangency?

    <p>m_{radius} imes m_{tangent} = -1</p> Signup and view all the answers

    In the equation of the circle, what does the term rac{E}{2} represent?

    <p>The y-coordinate of the center</p> Signup and view all the answers

    Which of the following properties holds true if two ratios are in proportion?

    <p>The sum of the products of the extremes equals the sum of the products of the means.</p> Signup and view all the answers

    Which statement is true about the properties of proportion?

    <p>If rac{w}{x} = rac{y}{z}, then w imes y = x imes z.</p> Signup and view all the answers

    How is the equation of a tangent line to a circle at the point of tangency written?

    <p>y - y_1 = m_{tangent}(x - x_1)</p> Signup and view all the answers

    What does the concept of ratio allow us to compare?

    <p>Two quantities with the same units</p> Signup and view all the answers

    What does Thales' theorem state about a line drawn parallel to one side of a triangle?

    <p>It creates similar triangles with the original triangle.</p> Signup and view all the answers

    What aspect of a circle is indicated by the center coordinates (a, b) in its standard form?

    <p>The location of the circle's center</p> Signup and view all the answers

    What occurs if a line is drawn parallel to one side of a triangle in terms of the other two sides?

    <p>The other two sides are divided proportionally.</p> Signup and view all the answers

    What is the compound angle formula for the sine of the difference of two angles?

    <p>$rac{ an(eta + eta)}{1 - an eta an eta}$</p> Signup and view all the answers

    How can the cosine of a sum be expressed using sine and cosine functions?

    <p>$ an(eta + eta) = rac{ an eta + an eta}{1 - an eta an eta}$</p> Signup and view all the answers

    Which of the following represents the correct formula for sine of double angle?

    <p>$ an(2 heta) = rac{2 an heta}{1 - an^2 heta}$</p> Signup and view all the answers

    What is the correct expression for the cosine of double angle in terms of sine?

    <p>$ an(2 heta) = rac{1 - 2rac{ an^2 heta}{ an^2 heta}}{1 + 2 an^2 heta}$</p> Signup and view all the answers

    What is the general method for solving trigonometric equations?

    <p>Identify exact values and their periodicities.</p> Signup and view all the answers

    Which equation correctly represents the sine of a sum?

    <p>$ an( heta + heta) = rac{ an heta + an heta}{1 - an^2 heta}$</p> Signup and view all the answers

    When deriving the cosine of a double angle, which identity is essential?

    <p>$ ext{sec}^2 heta = 1 + an^2 heta$</p> Signup and view all the answers

    Which statement best describes the co-function identity for sine?

    <p>$ an(90^ ext{circ} - heta) = rac{1}{ an heta}$</p> Signup and view all the answers

    What does the CAST diagram help determine when solving trigonometric equations?

    <p>Quadrants where trigonometric functions are positive or negative.</p> Signup and view all the answers

    What condition must be met for two triangles to be similar?

    <p>All pairs of corresponding angles must be equal.</p> Signup and view all the answers

    Which statement accurately describes the proportionality theorem as stated in the content?

    <p>The ratios of the lengths of the sides of two triangles are equal.</p> Signup and view all the answers

    How is the area of a triangle calculated?

    <p>Area = (1/2) × base × height</p> Signup and view all the answers

    What can be inferred if two triangles are equiangular?

    <p>They are similar triangles.</p> Signup and view all the answers

    What does the Pythagorean theorem state?

    <p>The square on the hypotenuse equals the sum of the squares of the adjacent sides.</p> Signup and view all the answers

    What does it mean when two triangles are similar?

    <p>Their corresponding angles are equal and their sides are in proportion.</p> Signup and view all the answers

    How is similarity in polygons determined?

    <p>All corresponding angles are equal and all corresponding sides are in proportion.</p> Signup and view all the answers

    Which of the following accurately represents the converse of the Pythagorean theorem?

    <p>If the triangle's sides satisfy the Pythagorean theorem, then it is a right-angled triangle.</p> Signup and view all the answers

    What is a necessary condition for a triangle to have equal areas when sharing a common base?

    <p>They must have equal altitudes.</p> Signup and view all the answers

    When should the Cosine Rule be applied in triangle problems?

    <p>When two sides and the included angle are known.</p> Signup and view all the answers

    What is the correct general solution for the equation \tan \theta = x?

    <p>\theta = \tan^{-1} x + k \cdot 180^\circ</p> Signup and view all the answers

    In the context of triangles, which rule allows the calculation of an area when no height is provided?

    <p>Area Rule</p> Signup and view all the answers

    Which of the following equations represents the Sine Rule?

    <p>\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}</p> Signup and view all the answers

    To find the height of a pole using trigonometric ratios, which of the following is used?

    <p>Combining the Sine Rule and the Tangent Ratio.</p> Signup and view all the answers

    What does the area of triangle ABC equal when using the Area Rule?

    <p>\frac{1}{2}ac \sin B</p> Signup and view all the answers

    If \sin \theta = x, which relationships can determine \theta?

    <p>\theta = 180^\circ - \sin^{-1} x + k \cdot 360^\circ</p> Signup and view all the answers

    Which condition indicates that the Sine Rule should be used?

    <p>Two angles and a side are known.</p> Signup and view all the answers

    What is the formula for the area of a parallelogram?

    <p>Area = base × height</p> Signup and view all the answers

    What is the name of the theorem that states if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally?

    <p>Basic Proportionality Theorem</p> Signup and view all the answers

    What is the formula for the area of a triangle?

    <p>Area = (base × height) / 2</p> Signup and view all the answers

    What is the relationship between the areas of two triangles with the same height?

    <p>The areas are proportional to their bases</p> Signup and view all the answers

    What is the formula for the area of a kite?

    <p>Area = (diagonal1 × diagonal2) / 2</p> Signup and view all the answers

    What is the relationship between the areas of two triangles with equal bases and between the same parallel lines?

    <p>The areas are equal</p> Signup and view all the answers

    What is the name of the theorem that states the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides?

    <p>Pythagoras' Theorem</p> Signup and view all the answers

    What is the formula for the area of a trapezoid?

    <p>Area = (base1 + base2) × height / 2</p> Signup and view all the answers

    What is the formula for the area of a rhombus?

    <p>Area = (diagonal1 × diagonal2) / 2</p> Signup and view all the answers

    What is the formula for the area of a square?

    <p>Area = side × side</p> Signup and view all the answers

    Given the equation of a circle in standard form: (x - a)^2 + (y - b)^2 = r^2, what is the coordinate of the center of the circle?

    <p>(a, b)</p> Signup and view all the answers

    What is the gradient of the tangent line to a circle at a point of tangency, if the gradient of the radius to that point is $m_{radius}$?

    <p>$-1/m_{radius}$</p> Signup and view all the answers

    If a line is drawn parallel to one side of a triangle, what relationship does it have with the other two sides?

    <p>It divides the other two sides proportionally.</p> Signup and view all the answers

    If the ratio of two quantities is 3:4, what is the ratio of their reciprocals?

    <p>4:3</p> Signup and view all the answers

    Which of the following is NOT a property of proportion?

    <p>Adding corresponding terms</p> Signup and view all the answers

    What is the simplified form of the equation: (x + D/2)^2 + (y + E/2)^2 = (D/2)^2 + (E/2)^2 - F?

    <p>(x - D/2)^2 + (y - E/2)^2 = F</p> Signup and view all the answers

    What is the radius of the circle represented by the equation: (x - 2)^2 + (y + 3)^2 = 16?

    <p>4</p> Signup and view all the answers

    What is the gradient of the radius of the circle (x - 3)^2 + (y + 2)^2 = 25 at the point of tangency (5, 1)?

    <p>3/2</p> Signup and view all the answers

    If the equation of a circle is x^2 + y^2 - 6x + 4y - 12 = 0, what is the equation of the tangent line at the point (5, 3)?

    <p>y - 3 = (3/2)(x - 5)</p> Signup and view all the answers

    Given a triangle ABC with a line DE parallel to BC and intersecting AB and AC at D and E respectively. If AD = 4, DB = 6, and AE = 5, what is the length of EC?

    <p>7.5</p> Signup and view all the answers

    Using the formula for (\cos(\alpha - \beta)), what is the simplified expression for (\cos(180^\circ - \theta))?

    <p>(-\cos(\theta))</p> Signup and view all the answers

    What is the formula for (\sin(\alpha + \beta)) in terms of sine and cosine of (\alpha) and (\beta)?

    <p>(\sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta))</p> Signup and view all the answers

    Which of the following correctly expresses (\cos(\alpha + \beta)) using the negative angle identity?

    <p>(\cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta))</p> Signup and view all the answers

    Given that triangles ABC and DEF have sides in proportion, what can we conclude about the triangles?

    <p>The triangles are similar.</p> Signup and view all the answers

    In the proof of the Pythagorean Theorem, why are triangles ABD and CBA similar?

    <p>They have three pairs of corresponding angles equal.</p> Signup and view all the answers

    If two triangles have equal heights and their bases are in the ratio 2:3, what is the ratio of their areas?

    <p>2:3</p> Signup and view all the answers

    Which of the following statements is true about similar polygons?

    <p>Corresponding angles of similar polygons are equal.</p> Signup and view all the answers

    What is the relationship between the areas of two triangles with the same base and between the same parallel lines?

    <p>The areas are equal.</p> Signup and view all the answers

    In the proof of the Pythagorean Theorem, what is the purpose of drawing AD perpendicular to BC?

    <p>All of the above.</p> Signup and view all the answers

    What is the converse of the Pythagorean Theorem?

    <p>If the square on one side of a triangle is equal to the sum of the squares on the other two sides, then the triangle is a right triangle.</p> Signup and view all the answers

    What is the importance of the Proportionality Theorem in the context of similar triangles?

    <p>It states that if two triangles are similar, their sides are in proportion.</p> Signup and view all the answers

    Which of the following conditions is NOT sufficient to prove that two triangles are similar?

    <p>Two pairs of corresponding angles are equal.</p> Signup and view all the answers

    What is the significance of the Pythagorean Theorem in geometry?

    <p>It is a fundamental theorem that relates the sides of a right-angled triangle.</p> Signup and view all the answers

    What can be concluded about triangles on the same base and equal in area?

    <p>They lie between parallel lines.</p> Signup and view all the answers

    What does the Proportion Theorem state regarding a line drawn parallel to one side of a triangle?

    <p>It divides the other two sides in the same ratio.</p> Signup and view all the answers

    According to the Mid-point Theorem, what is true about the line joining the midpoints of two sides of a triangle?

    <p>It is parallel to the third side.</p> Signup and view all the answers

    What must be true for two polygons to be considered similar?

    <p>Their angles must be equal and sides must be proportional.</p> Signup and view all the answers

    What does the equation of a circle with center at the origin state about any point on its circumference?

    <p>$x^2 + y^2 = r^2$</p> Signup and view all the answers

    Which statement is correct regarding equiangular triangles?

    <p>Their sides are in proportion.</p> Signup and view all the answers

    For a circle with center at point (a, b), how is the general equation expressed?

    <p>$(x - a)^2 + (y - b)^2 = r^2$</p> Signup and view all the answers

    What does the Converse of the Mid-point Theorem state?

    <p>A line from the midpoint of one side parallel to another side bisects the third side.</p> Signup and view all the answers

    What is a unique characteristic of a circle with respect to its symmetry?

    <p>It is symmetric about the origin.</p> Signup and view all the answers

    How are similar polygons defined?

    <p>Polygons that differ in size but have the same shape.</p> Signup and view all the answers

    What is one method to prove that two triangles are similar?

    <p>Show that two pairs of corresponding angles are equal.</p> Signup and view all the answers

    What is the purpose of completing the square in the equation of a circle?

    <p>To find the center and radius of the circle.</p> Signup and view all the answers

    Which of the following steps is NOT part of completing the square for the circulation equation?

    <p>Finding roots of the resulting quadratic equations.</p> Signup and view all the answers

    In the context of triangle areas, when triangles have the same height, what can be said about their bases?

    <p>Their areas are proportional to their bases.</p> Signup and view all the answers

    Why is the expression $OP = r$ significant in deriving the equation of a circle?

    <p>It relates the radius to a point in Cartesian coordinates.</p> Signup and view all the answers

    If two triangles are equiangular, what conclusion can be drawn?

    <p>Their corresponding sides are in proportion.</p> Signup and view all the answers

    Which equation corresponds to the standard form of a circle centered at (0, 0) with radius changes?

    <p>$x^2 + y^2 = r^2$</p> Signup and view all the answers

    In which scenario would you use the equation $(x - a)^2 + (y - b)^2 = r^2$?

    <p>When the center of the circle is at a point (a, b).</p> Signup and view all the answers

    Which of the following is a correct formula for the cosine of a double angle?

    <p>$\cos(2\alpha) = 2\cos^2\alpha - 1$</p> Signup and view all the answers

    What is the first step in deriving the double angle formula for sine?

    <p>Start with the sum formula for sine.</p> Signup and view all the answers

    Which of the following is NOT a valid formula for the cosine of a double angle?

    <p>$\cos(2\alpha) = \cos^2\alpha + \sin^2\alpha$</p> Signup and view all the answers

    Which of the following is the correct formula for the sine of a sum?

    <p>$\sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta$</p> Signup and view all the answers

    Which trigonometric rule is used when no perpendicular height is given in a triangle?

    <p>The Cosine Rule</p> Signup and view all the answers

    Which of the following is the correct formula for the sine of a difference?

    <p>$\sin(\alpha - \beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta$</p> Signup and view all the answers

    Which of the following is the formula for the cosine of a difference?

    <p>$\cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta$</p> Signup and view all the answers

    What is the purpose of using a CAST diagram when solving trigonometric equations?

    <p>To determine where the function is positive or negative.</p> Signup and view all the answers

    Which of the following is NOT a valid step in the general solution method for trigonometric equations?

    <p>Find angles in the interval ([0^\circ, 360^\circ]) that satisfy the equation and add multiples of the period, using the reference angle and the CAST diagram.</p> Signup and view all the answers

    Which of the following is the formula for the cosine of a sum?

    <p>$\cos(\alpha + \beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta$</p> Signup and view all the answers

    Which equation represents the correct general solution for \( an heta = x \)?

    <p>\( \theta = \tan^{-1} x + k \cdot 180^\circ \)</p> Signup and view all the answers

    When should the Sine Rule be applied in triangle problems?

    <p>When no right angle is given and two angles and a side are specified.</p> Signup and view all the answers

    Which formula correctly calculates the area of triangle ABC using the Sine Rule?

    <p>\( \text{Area} = \frac{1}{2}bc \sin A \)</p> Signup and view all the answers

    What conditions necessitate the use of the Cosine Rule?

    <p>When three sides are known, or two sides and the included angle are specified.</p> Signup and view all the answers

    Which of the following relationships is true when using the Sine Rule?

    <p>\( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \)</p> Signup and view all the answers

    What does the formula ( h = \frac{d \sin \alpha}{\sin \beta} \tan \beta ) represent?

    <p>The height of a pole.</p> Signup and view all the answers

    What is the correct generalization for the area of triangle ABC if only the lengths of two sides and the included angle are known?

    <p>\( \text{Area} = \frac{1}{2}ab \sin C \)</p> Signup and view all the answers

    According to the Sine Rule, what is ( BD ) expressed in terms of length ( b ), angle ( \theta ), ( \alpha ), and ( \beta )?

    <p>\( BD = \frac{b \sin \theta}{\sin(\beta + \theta)} \)</p> Signup and view all the answers

    If a circle has a center at (a, b) and radius r, what is the equation of the circle?

    <p>(x - a)^2 + (y - b)^2 = r^2</p> Signup and view all the answers

    What is the significance of completing the square in the equation of a circle?

    <p>To find the center and radius of the circle.</p> Signup and view all the answers

    A circle with center at the origin is symmetric about which of the following?

    <p>The x-axis, y-axis, and the origin</p> Signup and view all the answers

    What is the first step in completing the square to find the center and radius of a circle?

    <p>Group the x terms and the y terms</p> Signup and view all the answers

    What is the equation of a circle with center at the origin and radius r?

    <p>x^2 + y^2 = r^2</p> Signup and view all the answers

    What is the purpose of rewriting the equation of a circle in standard form?

    <p>To find the center and radius of the circle.</p> Signup and view all the answers

    What is the significance of the distance formula in the derivation of the equation of a circle?

    <p>It helps to derive the equation of the circle.</p> Signup and view all the answers

    What is the relationship between the equation of a circle with center at the origin and the equation of a circle with center at (a, b)?

    <p>They are different equations with the same form.</p> Signup and view all the answers

    If two triangles are similar, which of the following statements is true?

    <p>The corresponding sides are in proportion and the corresponding angles are equal.</p> Signup and view all the answers

    What is the formula for the area of a trapezoid?

    <p>Area = (base1 + base2) × height</p> Signup and view all the answers

    If a line is drawn parallel to one side of a triangle, what does it do to the other two sides?

    <p>It divides the other two sides proportionally.</p> Signup and view all the answers

    What is the formula for the area of a rhombus?

    <p>Area = (diagonal1 × diagonal2) / 2</p> Signup and view all the answers

    What is the name of the theorem that states if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally?

    <p>Thales' Theorem</p> Signup and view all the answers

    What is the formula for the area of a kite?

    <p>Area = (diagonal1 × diagonal2) / 2</p> Signup and view all the answers

    What is the relationship between the areas of two triangles with the same height?

    <p>The areas are proportional to their bases.</p> Signup and view all the answers

    What is the relationship between the areas of two triangles with equal bases and between the same parallel lines?

    <p>The areas are equal.</p> Signup and view all the answers

    Which of the following is a property of similar polygons?

    <p>The corresponding angles are equal, and the corresponding sides are in proportion.</p> Signup and view all the answers

    Which of the following is NOT a property of a kite?

    <p>The diagonals are parallel.</p> Signup and view all the answers

    If two triangles have equal heights and their bases are in the ratio 3:5, what is the ratio of their areas?

    <p>9:25</p> Signup and view all the answers

    In the figure below, if DE || BC and AD:DB = 2:3, what is the ratio of the area of triangle ADE to the area of triangle ABC?

    <p>4:9</p> Signup and view all the answers

    If two triangles are similar, which of the following statements is NOT true?

    <p>Their areas are equal.</p> Signup and view all the answers

    In the figure below, if PQRST is similar to ABCDE, what is the ratio of PQ to AB?

    <p>2:3</p> Signup and view all the answers

    If a line is drawn parallel to one side of a triangle, what does it do to the other two sides?

    <p>Divides them proportionally.</p> Signup and view all the answers

    What is the condition for two triangles to be similar?

    <p>They have equal corresponding angles and proportional corresponding sides.</p> Signup and view all the answers

    If two triangles are equiangular, what can be concluded about their corresponding sides?

    <p>They are proportional.</p> Signup and view all the answers

    What is the Mid-point Theorem?

    <p>A line joining the midpoints of two sides of a triangle is parallel to the third side and equal to half its length.</p> Signup and view all the answers

    What is the condition for two polygons to be similar?

    <p>They have the same shape but differ in size.</p> Signup and view all the answers

    What is the Converse of the Mid-point Theorem?

    <p>A line drawn from the midpoint of one side of a triangle parallel to another side bisects the third side.</p> Signup and view all the answers

    Which of the following is an alternate form of the cosine of a double angle formula?

    <p>$\cos(2\alpha) = 1 - 2\sin^2 \alpha$</p> Signup and view all the answers

    If $\sin(\alpha - eta) = rac{1}{2}$, what is the value of $\cos(\alpha + eta)$?

    <p>$-rac{1}{2}$</p> Signup and view all the answers

    What is the derivation of $\sin(2\alpha)$ using the sum formula for sine?

    <p>$\sin(2\alpha) = 2\sin \alpha \cos \alpha$</p> Signup and view all the answers

    What is the general solution method for trigonometric equations?

    <p>Simplify, Reference Angle, CAST Diagram, Restricted Values, General Solution, Check</p> Signup and view all the answers

    What is the formula for $\cos(\alpha - eta)$?

    <p>$\cos \alpha \cos eta + \sin \alpha \sin eta$</p> Signup and view all the answers

    What is the purpose of using a CAST diagram when solving trigonometric equations?

    <p>To determine where the function is positive or negative</p> Signup and view all the answers

    If $\cos(2\alpha) = rac{1}{2}$, what is the value of $\sin \alpha$?

    <p>$\pmrac{\sqrt{3}}{2}$</p> Signup and view all the answers

    What is the formula for $\sin(\alpha + eta)$?

    <p>$\sin \alpha \cos eta + \cos \alpha \sin eta$</p> Signup and view all the answers

    What is the derivation of $\cos(2\alpha)$ using the sum formula for cosine?

    <p>$\cos(2\alpha) = \cos^2 \alpha - \sin^2 \alpha$</p> Signup and view all the answers

    What is the general solution for the equation $\sin( heta) = rac{1}{2}$?

    <p>$ heta = 30^\circ + 360^\circ k$</p> Signup and view all the answers

    What is the correct expression for the radius of a circle in standard form using parameters D, E, and F?

    <p>$r = ext{sqrt}igg(igg(rac{D}{2}igg)^2 + igg(rac{E}{2}igg)^2 - Figg)$</p> Signup and view all the answers

    What is the gradient of the tangent line in relation to the gradient of the radius at the point of tangency?

    <p>$m_{tangent} = -m_{radius}$</p> Signup and view all the answers

    If the center of a circle is at (a, b), how would you express the gradient of the radius to a point of tangency (x1, y1)?

    <p>$m_{radius} = rac{y1 - b}{x1 - a}$</p> Signup and view all the answers

    What does the Basic Proportionality Theorem state about a triangle when a line is drawn parallel to one of its sides?

    <p>It divides the other two sides proportionally.</p> Signup and view all the answers

    What is the relationship expressed by the reciprocal proportion of two ratios $\frac{w}{x} = \frac{y}{z}$?

    <p>$\frac{x}{w} = \frac{y}{z}$</p> Signup and view all the answers

    To determine the equation of a tangent, what is the correct point-slope form using the point of tangency (x1, y1) and the tangent's gradient?

    <p>$y - y1 = m_{tangent}(x - x1)$</p> Signup and view all the answers

    If a line is drawn parallel to one side of a triangle, what is the effect on the segments it creates?

    <p>It divides the other two sides in proportion.</p> Signup and view all the answers

    What mathematical property does the statement $\frac{w}{x} = \frac{y}{z}$ exemplify?

    <p>Cross Multiplication.</p> Signup and view all the answers

    What key characteristic defines a diagonal ratio in relation to areas of similar polygons?

    <p>It dictates the relationship between the areas of triangles.</p> Signup and view all the answers

    If triangles ABC and DEF have corresponding sides in proportion, which angles must necessarily be equal?

    <p>Angle A and Angle E</p> Signup and view all the answers

    Which of the following conditions guarantees the similarity of two triangles?

    <p>All corresponding angles are equal</p> Signup and view all the answers

    For two triangles with equal height, how is their area related to their bases?

    <p>Areas are directly proportional to their bases</p> Signup and view all the answers

    What conclusion can be made about the triangles AGH and ABC if AG is equal to DE and AH is equal to DF?

    <p>Triangle AGH is similar to triangle DEF</p> Signup and view all the answers

    In proving the Pythagorean theorem, what relationship is established between the triangles ABD and CBA?

    <p>They are similar triangles</p> Signup and view all the answers

    Which of the following correctly states the Pythagorean theorem?

    <p>$BC^2 = AB^2 + AC^2$</p> Signup and view all the answers

    If triangles ABC and DEF are equiangular, what can be said about their corresponding sides?

    <p>Their corresponding sides are in proportion</p> Signup and view all the answers

    Which construction is essential for proving that \triangle AGH is similar to \triangle ABC?

    <p>Drawing a line parallel to BC from G</p> Signup and view all the answers

    What does it imply if two triangles' sides are not in proportion?

    <p>The triangles are not similar</p> Signup and view all the answers

    Using the distance formula and cosine rule, what is the first step in deriving the formula for (\cos(\alpha - \beta))?

    <p>Find the distance between two points on the unit circle, (K) and (L), using the distance formula.</p> Signup and view all the answers

    What is the primary purpose of using the negative angle identity in the derivation of the formula for (\cos(\alpha + \beta))?

    <p>To express (\cos(\alpha + \beta)) as (\cos(\alpha - (-eta))) and apply the cosine difference formula.</p> Signup and view all the answers

    During the construction of triangle AGH, GH is drawn parallel to which side, and why?

    <p>GH is parallel to BC to ensure corresponding angles are equal</p> Signup and view all the answers

    Which of the following correctly expresses the cosine of a sum in terms of the cosine of a difference?

    <p>(\cos(\alpha + \beta) = \cos(\alpha - (-eta)))</p> Signup and view all the answers

    Which trigonometric identity is directly used to derive the formula for (\sin(\alpha + \beta)) after applying the cosine difference formula?

    <p>Sine of a difference</p> Signup and view all the answers

    Which of the following steps is NOT required in the derivation of (\sin(\alpha + \beta)) using the cosine difference formula and other trigonometric identities?

    <p>Applying the Pythagorean identity to the expression obtained in step (b).</p> Signup and view all the answers

    What is the key difference between deriving (\cos(\alpha + \beta)) and deriving (\sin(\alpha + \beta))?

    <p>The derivation of (\cos(\alpha + \beta)) involves expressing the sum as a difference, while the derivation of (\sin(\alpha + \beta)) involves expressing the sum as a co-function.</p> Signup and view all the answers

    Which of the following is NOT a direct consequence of the derivation of the cosine of a difference formula?

    <p>The tangent of a difference and tangent of a sum formulas can be derived directly from the sine of a difference and cosine of a difference formulas.</p> Signup and view all the answers

    What is the essential role of the distance formula in the derivation of the cosine of a difference formula?

    <p>The distance formula is used to relate the distance between the two points on the unit circle to the cosine of the angle between their radii.</p> Signup and view all the answers

    What is the correct application of the Area Rule for calculating the area of a triangle?

    <p>It is used when no perpendicular height is given.</p> Signup and view all the answers

    Given \( an heta = x\), what is a general solution for \( heta\)?

    <p>\( heta = an^{-1} x + k \cdot 180^ imes \)</p> Signup and view all the answers

    In which scenario would you use the Cosine Rule?

    <p>When three sides are known.</p> Signup and view all the answers

    What formula is used to find the height of a pole given angle measurements and distance?

    <p>\( h = FB an eta \)</p> Signup and view all the answers

    Which of the following correctly describes the Sine Rule's application?

    <p>It is used when two angles and a side are known.</p> Signup and view all the answers

    For the equation \( a^2 = b^2 + c^2 - 2bc \cos A \, \) what does it represent?

    <p>The Cosine Rule applied to any triangle.</p> Signup and view all the answers

    What is the correct formulation to calculate the area of triangle ABC using the Sine Rule?

    <p>\( ext{Area} = rac{1}{2}bc \sin B \)</p> Signup and view all the answers

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