Question

Similar triangles: To indirectly measure the distance across a river, Madeline stands on one side of the river and uses sight-lines to a landmark on the opposite bank. Madeline 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.

Answer 1

PR = 288.75 feet

Explanation:

There are two triangles that we need to consider:
ΔPRE and ΔPOC

These triangles are similar since
m∠PRE = m∠POC = 90°
∠PRE ≅ ∠POC

∠RPE ≅ ∠OPC (common angle)

AA Similarity theorem..
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

∴ ΔPRE ≅ ΔPOC

If two triangles are similar, the ratios of the corresponding sides must be equal

⇒
`(PO)/(PR) = (OC)/(PE)\\\\` (1)

PO = PR + RO
PR is the width of the river that is to be determined
Let PR = x

Then
PO = x + 165

OC = 330, RE = 210

`(PO)/(PR) = (OC)/(PE)\\\\\Rightarrow (x + 165)/(165) = (330)/(210)\\\\`

Left side:

`(x+165)/(x) = (x)/(x) + (165)/(x) = 1 + (165)/(x)`

Simplify
`(330)/(210)` by dividing both numerator and denominator by 30

=>

`(330)/(210) = (11)/(7)\\\\`

Therefore

`(PO)/(PR) = (OC)/(PE)\\\\\Rightarrow 1 + (165)/(x) = (11)/(7)\\`

Subtract 1 from both sides:

`(165)/(x) = (11)/(7) - 1\\\\(165)/(x) = (4)/(7)\\\\`

Cross-multiply

`165 * 7 = 4x\\\\\Rightarrow 4x = 165 * 7\\\\\Rightarrow x = (165 * 7 )/(4) = 288.75\;feet`