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

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

Transcript text: To indirectly measure the distance across a river, Fatoumata stands on one side of the river and uses sight-lines to a landmark on the opposite bank. Fatoumata draws the diagram below to show the lengths and angles that she measured. Find $P R$, the distance across the river. Round your answer to the nearest foot.

Solution

Solution Steps

Step 1: Establish Similarity

Triangles $\triangle OEC$ and $\triangle REP$ are similar because they share angle $\angle E$ and they both have a right angle ($\angle O$ and $\angle R$ respectively). Therefore, corresponding sides have equal ratios.

Step 2: Set up a proportion

The ratio of corresponding sides $RE$ and $OE$ in $\triangle REP$ and $\triangle OEC$, respectively, is equal to the ratio of corresponding sides $PR$ and $OC$. This gives us the proportion:

$\frac{RE}{OE} = \frac{PR}{OC}$

Step 3: Substitute known values and solve

We know $RE = 180$ ft, $OE = RE + RO = 180 + 350 = 530$ ft, and $OC = 125$ ft. Substituting these values into our proportion gives:

$\frac{180}{530} = \frac{PR}{125}$

Now, solve for $PR$:

$PR = \frac{180 \times 125}{530} \approx 42.45$