Questions: Determine the length of PQ. If PQ needs to be broken into sections 11 m long, how many times will it need to be cut?

Solution Steps

Triangles ROS and RQP are similar because they share angle R, and angles ROS and RQP are corresponding angles formed by parallel lines and a transversal, thus congruent. Likewise, angles ORS and PRQ are congruent. Therefore, the triangles are similar by Angle-Angle similarity.

Since the triangles are similar, their corresponding sides are proportional. We have RS:RQ = RO:RP. We know RS = QS and QS:RQ = 1:2. Also, we know RO:RP = 1:2. Since RO = 110m, RP = 2 * RO.

RP = 2 * 110m = 220m. Since RP = RQ + QP, and RQ = RS + SQ = 2SQ, then RQ = 220m. Therefore PQ is the same length as RQ. PQ= 220m. Since PQ needs to be broken into 11m sections, it will need to be cut 220/11 = 20 times.

Final Answer