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 CB and AB. You must show all work and calculations to receive full credit. (5 points)

The first part of the question is missing. Here it is;

Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and cos X = 2.5/5.59

A)ΔXYZ and ΔACB are similar triangles because corresponding sides of both triangles are proportional while corresponding angles of both triangles are congruent.

B)AB = 11.18

CB = 10

Step-by-step explanation:

A) We are told that ΔXYZ was dilated by a scale factor of 2. This means that it was stretched by a factor of 2.

From the image of both triangles, we can see that;

∠Y = ∠C = 90º

Since ΔXYZ is a right angle triangle and it was only dilated by a factor of 2 to get ΔACB which is also a right angle triangles, then we can infer that;

∠X = ∠A

Now, in ΔXYZ, since ∠Y = 90°, then;

∠X + ∠Z = 90° (since sum of angles in a triangle is 180°)

Thus, ∠Z = 90° - ∠X

This means that ∠X and ∠Z are complementary angles.

Applying the same to ΔABC, we can say that;

∠B = 90° - ∠A

Thus, we can say that ∠A and ∠B are complementary angles.

Since they are similar triangles, we can say that; ∠Z = ∠B

From both triangles;

Since ∠X = ∠A ; ∠Z = ∠B and ∠Y = ∠C = 90º, it means we can conclude that ΔXYZ and ΔACB are similar triangles because corresponding sides of both triangles are proportional while corresponding angles of both triangles are congruent.

B) We are given;

cos ∠X = 2.5/5.59

In ΔXYZ;

cos ∠X = XY/XZ

Thus;

XY = 2.5 (the adjacent side)

XZ = 5.59 (hypotenuse)

From pythagoras theorem, we can find YZ

YZ = √(5.59² - 2.5²)

YZ = √24.9981

YZ ≈ 5

Now, since ΔXYZ was dilated by a factor of 2,it means that by corresponding sides;

CB/YZ = 2

Thus;

CB = 2YZ

CB = 2 × 5 =

CB = 10

Likewise;

AB/XZ = 2

AB = 2XZ

AB = 2 × 5.59

AB = 11.18

you didn't post any answer choices, so its not 100% clear what your teacher is looking for

however, log(15/7) is equivalent to log(15) - log(7)

i'm using the rule log(x/y) = log(x) - log(y)