Question

find vw.
(image of a triangle with points z, v, w on the base, zv = 18, and a line from y to w with segments xy = 44, y to the top of the other segment = 66. red marks indicate parallel lines. then vw = blank and a submit button.)

Explanation:

Step1: Identify Similar Triangles

The segments with arrows are parallel, so by the Basic Proportionality Theorem (Thales' theorem), the triangles are similar. Let \( VX \parallel YZ \), so \( \triangle WXV \sim \triangle WYZ \). The ratio of corresponding sides should be equal. The sides on the transversal are \( WX = 44 \), \( WY = 66 \), so the ratio of similarity is \( \frac{WX}{WY}=\frac{44}{66}=\frac{2}{3} \).

Step2: Set Up Proportion for Bases

Let \( VW = x \), then \( WZ = VW + ZV = x + 18 \). Since the triangles are similar, \( \frac{VW}{WZ}=\frac{WX}{WY} \), so \( \frac{x}{x + 18}=\frac{2}{3} \).

Step3: Solve the Proportion

Cross - multiply: \( 3x=2(x + 18) \). Expand the right side: \( 3x = 2x+36 \). Subtract \( 2x \) from both sides: \( 3x-2x=2x + 36-2x \), so \( x = 36 \).

Answer:

\( 36 \)