Question

in the map below, \\(\overline{pq}\\) is parallel to \\(\overline{st}\\).

image of a geometric figure with points p, q, r, s, t. pq is 48 km, pr is 36 km, rt is 81 km. pq is parallel to st.

what is the distance between s and t? if necessary, round to the nearest tenth.

\\(\bigcirc\\) 21.3 km
\\(\bigcirc\\) 60.8 km
\\(\bigcirc\\) 108 km
\\(\bigcirc\\) 117 km

Explanation:

Step1: Identify Similar Triangles

Since \(\overline{PQ} \parallel \overline{ST}\), \(\triangle PQR \sim \triangle STR\) (by AA similarity, as vertical angles at \(R\) are equal and alternate interior angles are equal due to parallel lines).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{PQ}{ST} = \frac{PR}{TR}\). We know \(PQ = 48\) km, \(PR = 36\) km, and \(TR = 81\) km. Let \(ST = x\). Then \(\frac{48}{x} = \frac{36}{81}\).

Step3: Solve for \(x\)

Cross - multiply: \(36x = 48\times81\). Calculate \(48\times81 = 3888\). Then \(x=\frac{3888}{36}=108\) km.

Answer:

108 km