please help :/ !

Question

Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and cos ∠X = 2 and 5 tenths over 5 and 59 hundredths. 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 AB. You must show all work and calculations to receive full credit. (5 points)

Answer 1

Let's start with part a. They mention the triangle was dilated by a scale factor of 2, which just means it kept the same proportions but the length of all the sides was doubled. If the triangles have the same proportions, then that means that the angles and ratios between the sides should be the same. Here's an example with trigonometry. Let's say AC is equal to x and BC is equal to y. Tan A would then be opp/adj, or y/x. If XYZ is dilated by a factor of 2, then XY = 2x and YZ = 2y So tan Y = 2y/2x, which is also equal to y/x. So all the trigonometric ratios in ABC and XYZ are equivalent to each other.

Now for part b. They say cos X = 2.5/5.59 (if I am reading your question correctly). Recall that the cosine of an angle is adjacent/hypotenuse. So the adjacent side to angle X, or XY, must be 2.5, and the hypotenuse, XZ, must be 5.59.

Now let's find the ABC equivalents to the side lengths we just found. XY = 2*AC, so AC = XY/2. This comes out to 2.5/2, or 1.25. AC = 1.25.

Next we solved for XZ the hypotenuse, which was 5.59. The hypotenuse of ABC was AB, so XZ = 2*AB. AB = XZ/2, or 5.59/2. AB = 2.8

Now double-check the answer. We know for any right triangle, the hypotenuse should be the longest side. So if we are correct, AB > AC. Is this true? Well AB = 2.8, and AC = 1.25. And 2.8>1.25. So it looks good!

If you want an extra challenge, try finding BC and YZ using the answers we found earlier.