Question

$\overline{vw} \parallel \overline{xz}$. find $yz$.
(image of a trapezoid-like figure with vertices v, w, y, and a line xz parallel to vw. lengths: vz = 20, wx = 24, xy = 12. need to find yz.)
$yz = \square$

Explanation:

Step1: Identify Similar Triangles

Since $\overline{VW} \parallel \overline{XZ}$, by the Basic Proportionality Theorem (Thales' theorem), $\triangle YXZ \sim \triangle YWV$ (similar triangles).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, $\frac{YX}{YW} = \frac{YZ}{YV}$. Let $YZ = x$, then $YV = YZ + ZV = x + 20$, and $YX = 12$, $YW = YX + XW = 12 + 24 = 36$. Substituting into the proportion: $\frac{12}{36} = \frac{x}{x + 20}$.

Step3: Solve the Proportion

Simplify $\frac{12}{36}$ to $\frac{1}{3}$. So, $\frac{1}{3} = \frac{x}{x + 20}$. Cross - multiply: $x + 20 = 3x$. Subtract $x$ from both sides: $20 = 2x$. Divide both sides by 2: $x = 10$.

Answer:

10