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

The dilation of a triangle to another, by a scale factor greater than 1, means the resulting triangle will have greater side lengths than the original triangle. However, both triangles will be similar.

The special relationship between the trigonometry ratios of both triangles are:

Side lengths of AC and AB are:

and

The given parameters can be represented as:

 --- the scale factor

--- the similar triangles

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From  

We can calculate the side lengths of

The cosine of an angle is:

Using , we have:

Given that:

It means:

Using Pythagoras theorem, we have:

Collect like terms

Take square roots

Using the following trigonometry formulas:

The trigonometry ratios of the angles in  are:

Because , the following relationships exist:

To calculate AC and CB:

We have:

The scale ratio of to is 2:

This means that:

Sides of   = 2 * corresponding sides of

i.e.

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