Question

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

Answer 1

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:

$\tan X =\tan A= \frac{5.0}{2.5}$              $\tan Z = \tan B =\frac{1}{2}$

$\sin X = \sin A = \frac{5.0}{5.59}$              $\sin Z = \sin B =\frac{2.5}{5.59}$

$\cos X = \cos A = \frac{2.5}{5.59}$             $\cos Z =\cos B= \frac{5.0}{5.59}$

Side lengths of AC and AB are:

$AC=5$ and $AB = 11.18$

The given parameters can be represented as:

$\cos X = \frac{2.5}{5.59}$

$r = 2$  --- the scale factor

$\triangle XYZ \sim \triangle ACB$ --- the similar triangles

Also, see attachment

From $\cos X = \frac{2.5}{5.59}$  

We can calculate the side lengths of $\triangle XYZ$

The cosine of an angle is:

$\cos(\theta) =\frac{Adjacent}{Hypotenuse}$

Using $\triangle XYZ$, we have:

$\cos X = \frac{XY}{XZ}$

Given that:

$\cos X = \frac{2.5}{5.59}$

It means:

$XY = 2.5$

$XZ = 5.59$

Using Pythagoras theorem, we have:

$XZ^2 =XY^2 + YZ^2$

$5.59^2 =2.5^2 + YZ^2$

$31.2481 =6.25+ YZ^2$

Collect like terms

$YZ^2 = 31.2481 - 6.25$

$YZ^2 = 24.9981$

Take square roots

$YZ = 5.0$

Using the following trigonometry formulas:

$\tan(\theta) = \frac{Opposite}{Adjacent}$

$\sin(\theta) = \frac{Opposite}{Hypotenuse}$

$\cos(\theta) = \frac{Adjacent}{Hypotenuse}$

The trigonometry ratios of the angles in  are:

$\tan X = \frac{5.0}{2.5}$              $\tan Z = \frac{1}{2}$

$\sin X = \frac{5.0}{5.59}$             $\sin Z = \frac{2.5}{5.59}$

$\cos X = \frac{2.5}{5.59}$             $\cos Z = \frac{5.0}{5.59}$

Because $\triangle XYZ \sim \triangle ACB$, the following relationships exist:

$\tan X =\tan A= \frac{5.0}{2.5}$              $\tan Z = \tan B =\frac{1}{2}$

$\sin X = \sin A = \frac{5.0}{5.59}$              $\sin Z = \sin B =\frac{2.5}{5.59}$

$\cos X = \cos A = \frac{2.5}{5.59}$             $\cos Z =\cos B= \frac{5.0}{5.59}$

To calculate AC and CB:

We have:

$XY = 2.5$

$YZ = 5.0$

$XZ = 5.59$

The scale ratio of $\triangle XYZ$ to $\triangle ACB$ is 2:

This means that:

Sides of  $\triangle ACB$ = 2 * corresponding sides of $\triangle XYZ$

i.e.

$AC=2 * XY$

$AC=2 * 2.5$

$AC=5$

$AB = 2 * XZ$

$AB = 2 * 5.59$

$AB = 11.18$

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