Question

to indirectly measure the distance across a lake, kayden makes use of a couple landmarks at points p and q. he measures or, rp, and rs as marked. find the distance across the lake (pq), rounding your answer to the nearest hundredth of a meter. (diagram is not to scale.)

Explanation:

Step1: Identify similar triangles

Since $\angle{OPR}=\angle{ORS} = 90^{\circ}$ and $\angle{O}$ is common to both $\triangle{OPR}$ and $\triangle{ORS}$, the two triangles $\triangle{OPR}$ and $\triangle{ORS}$ are similar. Also, $\triangle{OPQ}$ and $\triangle{ORS}$ are similar because of the parallel - like structure (the right - angled nature and the common angle at $O$). The ratio of corresponding sides of similar triangles is equal. That is, $\frac{PQ}{RS}=\frac{PR}{OR}$.

Step2: Substitute the given values

We are given that $PR = 100$ m, $OR=130$ m, and $RS = 129.2$ m. Substituting these values into the proportion $\frac{PQ}{RS}=\frac{PR}{OR}$, we get $PQ=\frac{PR\times RS}{OR}$.

Step3: Calculate the value of PQ

$PQ=\frac{100\times129.2}{130}=\frac{12920}{130}\approx99.38$ m.

Answer:

$99.38$ m