Question

mathematical connections find the value of x for which \\( \overline{pq} \parallel \overline{rs} \\). triangle with segments: pq side labeled 3x + 5, pt side labeled 2x + 4, ts segment 5, tr segment 7, and rs segment parallel to pq

Explanation:

Step1: Apply the Basic Proportionality Theorem (Thales' Theorem)

If \( PQ\parallel RS\), then \(\frac{TR}{RQ}=\frac{TS}{SP}\).
Substituting the given values: \(\frac{7}{3x + 5}=\frac{5}{2x+4}\).

Step2: Cross - multiply

Cross - multiplying gives \(7(2x + 4)=5(3x + 5)\).
Expanding both sides: \(14x+28 = 15x+25\).

Step3: Solve for \(x\)

Subtract \(14x\) from both sides: \(28=x + 25\).
Subtract \(25\) from both sides: \(x=28 - 25\).

Answer:

\(x = 3\)