Question

To indirectly measure the distance across a river, Jonathan stands on one side of the river and uses sight-lines to a landmark on the opposite bank. Jonathan draws the diagram below to show the lengths and angles that he measured. Find P, RPR, the distance across the river. Round your answer to the nearest foot.

Answer 1

The distance PR across the river is approximately 144 feet.

To find the distance PR across the river, we can use the law of sines in right triangle POC.

The law of sines states:

`\[(\sin A)/(a) = (\sin B)/(b) = (\sin C)/(c)\]`

In triangle POC:

`\[(\sin(\angle POC))/(OP) = (\sin(\angle OCP))/(OC)\]`

Given that
`\(\angle POC\)` is a right angle,
`\(\sin(\angle POC) = \sin(90^\circ) = 1\)`, and we can rewrite the equation as:

`\[(1)/(OP) = (\sin(\angle OCP))/(OC)\]`

Now, substitute the known values:

`\[(1)/(OP) = ((RE)/(RO))/(OC)\]`

`\[(1)/(OP) = ((255)/(115))/(320)\]`

Now, solve for OP:

`\[OP = (320 * 115)/(255)\]`

`\[OP \approx 144 \text{ ft}\]`

So, the distance PR across the river is approximately 144 feet.