Question

Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and tan ∠X = 5 over 2 and 5 tenths.

Answer 1

When a triangle is dilated by a scale factor greater than 1, the resulting triangle will be bigger than the original triangle. However, both triangles will be similar, but not congruent

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

$\tan X =\tan A = \frac{5}{2.5}$       $\tan Z =\tan B =\frac{1}{2}$

$\sin X = \sin A = \frac{5}{5.6}$        $\sin Z = \sin B =\frac{2.5}{5.6}$

$\cos X = \cos A = \frac{2.5}{5.6}$        $\cos Z = \cos B= \frac{5}{5.6}$

$AC = 5$ and $CB = 10$

The given parameters can be represented as:

$\tan X =\frac{5}{2.5}$

$k = 2$ --- the scale factor

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

From $\tan X =\frac{5}{2.5}$

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

The tangent of an angle is:

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

Using $\triangle XYZ$, we have:

$\tan(X) = \frac{YZ}{XY}$

Given that:

$\tan X =\frac{5}{2.5}$

It means that

$YZ = 5$

$XY = 2.5$

Using Pythagoras theorem, we have:

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

$XZ^2 = 5^2 + 2.5^2$

$XZ^2 = 31.25$

Take square roots

$XZ = \sqrt{31.25$

$XZ = 5.6$

Using the following trigonometry formulas:

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

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

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

The trigonometry ratios of angles in $\triangle XYZ$ are:

$\tan X =\frac{5}{2.5}$       $\tan Z =\frac{1}{2}$

$\sin X = \frac{5}{5.6}$        $\sin Z = \frac{2.5}{5.6}$

$\cos X = \frac{2.5}{5.6}$        $\cos Z = \frac{5}{5.6}$

Because $\triangle XYZ \sim \triangle ACB$ then:

$\tan X =\tan A = \frac{5}{2.5}$       $\tan Z =\tan B =\frac{1}{2}$

$\sin X = \sin A = \frac{5}{5.6}$        $\sin Z = \sin B =\frac{2.5}{5.6}$

$\cos X = \cos A = \frac{2.5}{5.6}$        $\cos Z = \cos B= \frac{5}{5.6}$

To calculate AC and CB, we do the following:

In previous section

$YZ = 5$

$XY = 2.5$

$XZ = 5.6$

$\triangle XYZ$ was dilated by scale factor 2 to get $\triangle ACB$ means that:

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

i.e.

$AC = 2 * XY$

$AC = 2 * 2.5$

$AC = 5$

$CB = 2 * YZ$

$CB = 2 * 5$

$CB = 10$

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