Question

to indirectly measure the distance across a river, arun stands on one side of the river and uses sight - lines to a landmark on the opposite bank. arun draws the diagram below to show the lengths and angles that he measured. find pr, the distance across the river. round your answer to the nearest foot.

Explanation:

Step1: Use similarity of triangles

Since \(\angle PRQ=\angle QOC = 90^{\circ}\) and \(\angle PQR=\angle OQC\) (common angle), \(\triangle PQR\sim\triangle OQC\) by AA (angle - angle) similarity criterion.
For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{PR}{OQ}=\frac{RE}{OC}\).
Let \(PR = x\). We know \(OQ = 100\) ft, \(RE=145\) ft, and \(OC = 225\) ft.

Step2: Substitute values into the proportion

Substitute the values into \(\frac{x}{100}=\frac{145}{225}\).
Cross - multiply: \(225x=145\times100\).
\(225x = 14500\).
Solve for \(x\): \(x=\frac{14500}{225}=\frac{580}{9}\approx64.44\).

Answer:

\(64\) feet