Question

in the diagram below, ( pq = 42 ), ( pr = 24 ), and ( qn = 21 ). find the length of ( overline{ro} ). round your answer to the nearest tenth if necessary.
answer attempt 1 out of 2
( ro=)

Explanation:

Step1: Prove similarity of triangles

Since \(\angle PQR=\angle N\) (given) and \(\angle P=\angle P\) (common angle), \(\triangle PQR\sim\triangle PNO\) by the AA (Angle - Angle) similarity criterion.

Step2: Set up proportion

For similar triangles \(\triangle PQR\) and \(\triangle PNO\), we have the proportion \(\frac{PQ}{PN}=\frac{PR}{PO}\). First, find \(PN = PQ+QN=42 + 21=63\). Let \(RO=x\), then \(PO=PR + RO=24+x\). Substituting the values into the proportion: \(\frac{42}{63}=\frac{24}{24 + x}\).

Step3: Cross - multiply and solve for \(x\)

Cross - multiplying gives \(42(24 + x)=63\times24\).
Expand: \(42\times24+42x=63\times24\).
\(1008+42x = 1512\).
Subtract \(1008\) from both sides: \(42x=1512 - 1008\).
\(42x=504\).
Divide both sides by \(42\): \(x = 12\).

Answer:

\(12\)