Question

a surveyor wants to use similar triangles to determine the distance across a lake as shown at the
right.
a) are the two triangles in the figure similar? justify the answer.
b) what is the distance d across the lake?
b) what is the distance d across the lake? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the distance across the lake is  ft. (simplify your answer.)
b. the distance cannot be determined because the triangles are not similar.
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Explanation:

Step1: Check similarity

Since two angles of one triangle are equal to two angles of the other triangle (both have a \(75^{\circ}\) angle and the vertical angles are equal), by the AA (Angle - Angle) similarity criterion, the two triangles are similar.

Step2: Set up proportion

For similar triangles, the ratios of corresponding sides are equal. Let \(d\) be the distance across the lake. We have the proportion \(\frac{100}{150}=\frac{20}{d}\).
Cross - multiply: \(100d = 150\times20\).

Step3: Solve for \(d\)

\(100d=3000\), then \(d=\frac{3000}{100}=30\).

Answer:

a) Yes, the two triangles are similar (by AA similarity criterion).
b) A. The distance across the lake is \(30\) ft.