Question

if om = 61, mn = 54, on = 81, pq = 27, and rq = 40.5, find the perimeter of △pqr. round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Explanation:

Step1: Identify Similar Triangles

Triangles \( \triangle OMN \) and \( \triangle PQR \) have the same angle measures ( \( 42^\circ \), \( 49^\circ \), \( 89^\circ \) ), so they are similar by AA similarity.

Step2: Find the Scale Factor

The ratio of corresponding sides: \( \frac{PQ}{MN} = \frac{27}{54} = \frac{1}{2} \), \( \frac{RQ}{ON} = \frac{40.5}{81} = \frac{1}{2} \). So scale factor \( k = \frac{1}{2} \).

Step3: Find \( PR \)

Corresponding to \( OM = 61 \), \( PR = OM \times k = 61 \times \frac{1}{2} = 30.5 \).

Step4: Calculate Perimeter of \( \triangle PQR \)

Perimeter \( = PQ + RQ + PR = 27 + 40.5 + 30.5 \).

Answer:

\( 98.0 \)