Question

in △opq, \\(\overline{op} \parallel \overline{rs}\\). given that \\(qr = 10\\), \\(sp = 15\\), and \\(qs = 25\\), find \\(ro\\).

Explanation:

Step1: Apply Basic Proportionality Theorem

Since \( \overline{OP} \parallel \overline{RS} \) in \( \triangle OPQ \), by the Basic Proportionality Theorem (Thales' theorem), we have \( \frac{QR}{RO}=\frac{QS}{SP} \).

Step2: Substitute the given values

We know \( QR = 10 \), \( SP = 15 \), and \( QS = 25 \). Substituting these values into the proportion \( \frac{10}{RO}=\frac{25}{15} \).

Step3: Solve for \( RO \)

Cross - multiply to get \( 25\times RO=10\times15 \). Then \( 25RO = 150 \). Divide both sides by 25: \( RO=\frac{150}{25}=6 \).

Answer:

6