Question

question
to indirectly measure the distance across a river, kiran stands on one side of the river and uses sight - lines to a landmark on the opposite
bank. kiran 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.
(diagram is not to scale.)
answer attempt 1 out of 2
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Explanation:

Step1: Identify similar triangles

Since \(\angle O=\angle R = 90^{\circ}\) and \(\angle CPO=\angle EPR\) (common - angle), \(\triangle CPO\sim\triangle EPR\) by the AA (angle - angle) similarity criterion.

Step2: Set up the proportion

For similar triangles \(\triangle CPO\) and \(\triangle EPR\), the ratios of corresponding sides are equal. That is \(\frac{CP}{EP}=\frac{OP}{RP}=\frac{CO}{RE}\). We use \(\frac{OP}{RP}=\frac{CO}{RE}\). Let \(PR = x\). We know that \(OP=160\) ft, \(CO = 305\) ft, and \(RE = 195\) ft.
The proportion is \(\frac{160 + x}{x}=\frac{305}{195}\) (using \(CP=CO + OP=305 + 160\) and \(EP = RE\)). Cross - multiply: \(195(160 + x)=305x\).
Expand: \(195\times160+195x=305x\).
\(31200+195x = 305x\).

Step3: Solve for \(x\)

Subtract \(195x\) from both sides: \(31200=305x - 195x\).
\(31200 = 110x\).
\(x=\frac{31200}{110}\approx284\)

Answer:

\(284\)