Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and sin ∠X =5/5.59 .

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. 

Part B: Explain how to find the measures of segments CB and AB. You must show all work and calculations to receive full credit. 

Part A:

??XYZ∼?ACB? by SAS Similarity Theorem

Part B:

The measures of the remaining segments are:

?XY≈2.50? and ?CB≈5.00? through Pythagorean Theorem

Part A:

Given that, ?sin∠X=5/5.9??, ??XYZ? can be drawn where segment ?ZY? as the opposite side of the triangle and segment ?ZX? as the hypotenuse of the triangle.

When ??XYZ? was dilated by a scale factor of ?2? , it was mapped onto ??ABC?. Thus the new sine ratio of the dilated triangle is ?sin∠A=10/11.18?? .

Notice that the sine ratio of angle ?X? is proportional to

the sine ratio of angle ?A?, that is:

?5/5.59?=10/11.18??. Using Inverse Sine Function, we may solve for the values of ?∠X? and ?∠A? .

?∠X=sin^−1 5/5.59? ≈63.44??

?∠A=sin^−1 10/11.18? ≈63.44??

Thus, it can be stated that ??XYZ∼?ACB? by SAS Similarity Theorem.

"If two sides in one triangle are proportional to the two sides in another triangle and the included angle in both are congruent, then the two triangles are similar.

Part B:

The measures of the remaining sides can be found through Pythagorean Theorem:

?x^2+y^2=r^2?

for ??XYZ? :

?5^2+XY^2=5.59^2?

?XY= under root 5.59^2−5^2

?≈2.50?

for ??ACB? :

?10^2+AC^2=11.18^2?

?A=under root11.18^2−10^2

?≈5.00?

PFA