Question

1. (5 points) to measure the distance across a river, henry stands on one side of the river and uses sight - lines to a landmark on the opposite bank. find the distance across the river. round your answer to the nearest foot.

Explanation:

Step1: Determine similar triangles

Since \(\angle O=\angle R = 90^{\circ}\) and \(\angle C=\angle E\) (by AA similarity criterion), \(\triangle OCP\sim\triangle REP\).

Step2: Set up proportion

For similar triangles \(\triangle OCP\) and \(\triangle REP\), the ratios of corresponding sides are equal. So, \(\frac{OP}{RP}=\frac{OC}{RE}\). Let \(RP = x\) (the distance across the river we want to find). We know \(OP = 165\), \(OC = 330\), and \(RE = 210\). Substituting into the proportion: \(\frac{165}{x}=\frac{330}{210}\).

Step3: Solve the proportion

Cross - multiply: \(330x=165\times210\). Then \(x=\frac{165\times210}{330}\). Simplify: \(x = 105\).

Answer:

\(105\) ft