Question

triangle similarity theorems
if $overline{cd} \parallel \overline{xz}$ and $cx = 5$ units, what is $dz$?
5 units
4 units
2 units
3 units
(image of triangle with points y, d, z, c, x, and side lengths 25, 20, 15, 18)

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{CD} \parallel \overline{XZ} \), by the Basic Proportionality Theorem (Thales' theorem), \( \triangle YCD \sim \triangle YXZ \). So, the ratios of corresponding sides are equal: \( \frac{YC}{YX}=\frac{YD}{YZ}=\frac{CD}{XZ} \).

First, find \( YX \): \( YX = YC + CX \). Given \( YC = 25 \) and \( CX = 5 \), so \( YX = 25 + 5 = 30 \).

Find \( YZ \): Let \( DZ = x \), then \( YZ = YD + DZ = 20 + x \).

Step2: Set Up Proportion for Sides

From similarity, \( \frac{YC}{YX}=\frac{YD}{YZ} \). Substitute values: \( \frac{25}{30}=\frac{20}{20 + x} \).

Simplify \( \frac{25}{30} \) to \( \frac{5}{6} \). So, \( \frac{5}{6}=\frac{20}{20 + x} \).

Cross - multiply: \( 5(20 + x)=6\times20 \).

Step3: Solve for \( x \) (DZ)

Expand left side: \( 100 + 5x = 120 \).

Subtract 100 from both sides: \( 5x = 120 - 100 = 20 \).

Divide both sides by 5: \( x=\frac{20}{5}=4 \).

Answer:

4 units