Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and tan ∠X = 5 over 2 and 5 tenths. Triangles XYZ and ACB; angles Y and C both measure 90 degrees, angles A and X are congruent.

Part A: Use complete sentences to explain the special relationship between the trigonometric ratios of triangles XYZ and ACB. 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)

cosx=2.5/5.59

   "   =.4472

 "x"  =63.434°; so "a" is=.

When talking about triangles and a scale; the angles remain=. In this case; "a" and "x"; y=c; z=b.

The sides are 2x or x/2.

Part A: Dilation proportionately changes side lengths, but it does not change the angle sizes. Therefore, the triangles are similar triangles.

Dilation means the triangle ΔXYZ was stretched by a factor of "2".

m∠Y = m∠C = 90º ← given

ΔXYZ and ΔACB are the right triangles.

∠X ≅ ∠A ← given

∠X and ∠Z are complementary angles

m∠Z = 90° - m∠X

∠A and ∠B are complementary angles

m∠B = 90° - m∠A

Therefore, ∠Z ≅ ∠B

ΔXYZ ∼ ΔACB → similar triangles: corresponding sides are proportional, corresponding angles are congruent

sin∠X = 5/5.59 ← given

sin∠X = YZ/XY

YZ = 5 (units) → leg in ΔXYZ

XZ = 5.59 (units) → hypotenuse in ΔXYZ

Calculating the length of leg XY:

XY = √((5.59)2 - 52) ≅ 2.4996 (units)

CB/YZ = 2 ← given

CD = 2YZ = 2 • 5 = 10 (units)

AC/XY = 2 ← given

AC = 2XY ≅ 2 • 2.4996 ≅ 4.999 (units)

Step-by-step explanation: