Question

To indirectly measure the distance across a lake, Jeremiah makes use of a couplelandmarks at points O and P. He measures NR, RP, and Q R as marked. Find thedistance across the lake (OP), rounding your answer to the nearest hundredth of ameter.

Answer 1

Notice that triangles NPQ and NRQ are similar, then:

$\frac{OP}{QR}=\frac{NP}{NR}.$

Now, notice that:

$PN=NR+RP.$

Therefore:

$\frac{OP}{QR}=\frac{NR+RP}{NR}.$

Multiplying the above result by QR we get:

$\begin{gathered} \frac{OP}{QR}\times QR=\frac{NR+RP}{NR}\times QR, \\ OP=\frac{NR+RP}{NR}\times QR. \end{gathered}$

Substituting NR=125m, RP=65m, and QR=106.75m we get:

$OP=\frac{125m+65m}{125m}\times106.75m.$

Simplifying the above result we get:

$\begin{gathered} OP=\frac{190m}{125m}\times106.75m \\ =162.26m. \end{gathered}$

Answer:

$OP=162.26m.$