Question

wx || vv, find uv.
(there is a diagram with segments and lengths: 25, 10, 50, and a triangle-like figure with points u, a, x, w, v. the final part has \uv = \)

Explanation:

Step1: Identify Similar Triangles

Since \( WX \parallel VY \), triangles \( UVY \) and \( UWX \) are similar by the Basic Proportionality Theorem (Thales' theorem).

Step2: Set Up Proportion

Let \( UV = x \). Then \( UW = UV + VW = x + 50 \). The ratio of corresponding sides in similar triangles is equal. So, \(\frac{UV}{UW}=\frac{VY}{WX}\). Given \( VY = UV \) (wait, no, from the diagram, \( VY \) length: \( UX = 25 \), \( AX = 10 \)? Wait, maybe \( VY \) is \( 25 - 10 = 15 \)? Wait, no, let's re - examine. Wait, the length from \( U \) to \( X \) is 25, and from \( A \) to \( X \) is 10, so \( U \) to \( A \) is \( 25 - 10 = 15 \). So \( VY = 15 \) and \( WX = 25 \), and \( VW = 50 \). Let \( UV = x \), then \( UW=x + 50\). Since \( \triangle UVY\sim\triangle UWX \), \(\frac{x}{x + 50}=\frac{15}{25}\) (simplify \(\frac{15}{25}=\frac{3}{5}\)).

Step3: Solve the Proportion

Cross - multiply: \( 5x=3(x + 50) \)
Expand: \( 5x=3x+150 \)
Subtract \( 3x \) from both sides: \( 5x-3x=3x + 150-3x\)
\( 2x = 150 \)
Divide by 2: \( x=\frac{150}{2}=75 \)

Answer:

75