Question

δabc ~ δxyz. find side length x. (there is a diagram of two right triangles, δabc with right angle at c, hypotenuse ab = 25, and side ac = x; δxyz with right angle at z, legs xz = 6, zy = 8, hypotenuse xy = 10.)

Explanation:

Step1: Identify Similar Triangles

Triangles \( \triangle ABC \) and \( \triangle XYZ \) are right - angled and similar (by AA similarity criterion, as both have a right angle and the angles are proportional). For similar triangles, the ratios of corresponding sides are equal.
First, find the ratio of the hypotenuses. In \( \triangle XYZ \), hypotenuse \( XY = 10 \), and in \( \triangle ABC \), hypotenuse \( AB = 25 \). The ratio of hypotenuses \( \frac{AB}{XY}=\frac{25}{10}=\frac{5}{2} \).
Also, in \( \triangle XYZ \), one leg \( YZ = 8 \), and in \( \triangle ABC \), the corresponding leg (let's say \( BC \)) and in \( \triangle XYZ \), leg \( XZ = 6 \), and in \( \triangle ABC \), the corresponding leg (let's say \( AC=x \)).

Step2: Set Up Proportion for Corresponding Sides

Since the triangles are similar, the ratio of corresponding legs is equal to the ratio of hypotenuses.
We know that in \( \triangle XYZ \), the leg \( XZ = 6 \) and in \( \triangle ABC \), the corresponding leg is \( AC=x \), and the ratio of hypotenuses is \( \frac{5}{2} \).
So we set up the proportion: \( \frac{x}{6}=\frac{25}{10} \) (because the ratio of corresponding sides of similar triangles is equal).
Simplify the right - hand side: \( \frac{25}{10}=\frac{5}{2} \).
Now, solve for \( x \): \( x = 6\times\frac{5}{2} \).
Calculate \( 6\times\frac{5}{2} \): \( 6\div2 = 3 \), and \( 3\times5=15 \).

Answer:

15