Question

$\overline{pq} \parallel \overline{rt}$. find $ps$.
(image of a triangle with points q, r, s, t, p; qr = 18, rs = 12, tp = 42)

Explanation:

Step1: Apply Basic Proportionality Theorem

Since \( \overline{PQ} \parallel \overline{RT} \), by the Basic Proportionality Theorem (Thales' theorem), we have \( \frac{SR}{RQ}=\frac{ST}{TP} \). Let \( ST = x \) and \( PS=ST + TP=x + 42 \). We know \( SR = 12 \), \( RQ = 18 \), and \( TP = 42 \). So, \( \frac{12}{18}=\frac{x}{42} \).

Step2: Solve for \( x \)

Cross - multiply: \( 18x=12\times42 \). Calculate \( 12\times42 = 504 \). Then \( x=\frac{504}{18}=28 \).

Step3: Find \( PS \)

Now, \( PS=ST + TP \). We found \( ST = 28 \) and \( TP = 42 \), so \( PS=28 + 42=70 \).

Answer:

\( 70 \)