Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and tan ∠X = 5 over 2 and 5 tenths. Part A: Use complete sentences to explain the special relationship between the trigonometric ratios of triangles XYZ and ABC. You must show all work and calculations to receive full credit. (5 points) Part B: Explain how to find the measures of segments AC and CB. You must show all work and calculations to receive full credit. (5 points) Triangles XYZ and ACB; angles Y and C both measure 90 degrees, angles A and X are congruent.

Step-by-step explanation:

Part A:

Trigonometric ratios are just that: ratios of side lengths. Since dilation changes all side lengths by the same dilation factor, the ratios of side lengths remain unchanged. Hence all trig functions of one triangle are precisely equal to the same trig functions of any similar triangle, regardless of the dilation factor.

Example*

  tan(X) = YZ/YX = 5/2.5 = 2

  tan(A) = CB/CA = (2·5)/(2·2.5) = 2

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Part B:

AC is the segment of ∆ACB that corresponds to segment XY of ∆XYZ. Its length is the length of XY, given as 2.5, multiplied by the scale factor (2). Hence ...

  AC = 2XY = 2(2.5)

  AC = 5

Similarly, ...

  AB = 2XZ = 2(5)

  AB = 10

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* SOH CAH TOA tells us that the Tangent is the ratio of the Opposite to the Adjacent segments in the right triangle. Since the ratio is given as 5/2.5, we presume these are the raw lengths of the respective line segments: YZ/XY.

11) obtuse

13)right

15) acute