Question

to indirectly measure the distance across a lake, katherine makes use of a couple landmarks at points q and r. she measures ps, sq, and st as marked. find the distance across the lake (qr), rounding your answer to the nearest hundredth of a meter. (diagram is not to scale.)

Explanation:

Step1: Identify Similar Triangles

Triangles \( \triangle PS T \) and \( \triangle PQ R \) are similar (by AA similarity, as both have a right angle and share \( \angle P \)).

Step2: Set Up Proportion

Let \( QR = x \). The proportion is \( \frac{ST}{QR}=\frac{PS}{PQ} \). We know \( ST = 147.6 \) m, \( PS = 180 \) m, \( SQ = 70 \) m, so \( PQ=PS + SQ=180 + 70 = 250 \) m. Substituting values: \( \frac{147.6}{x}=\frac{180}{250} \).

Step3: Solve for \( x \)

Cross - multiply: \( 180x=147.6\times250 \). Then \( x=\frac{147.6\times250}{180} \). Calculate \( 147.6\times250 = 36900 \), then \( x=\frac{36900}{180}=205.00 \) (rounded to nearest hundredth).

Answer:

\( 205.00 \) meters