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?
a) are the two triangles similar? justify the answer. choose the correct answer below.
a. no. the triangles are not similar because dilation is not considered a similarity transformation. only rigid motions are similarity transformations.
b. no. the triangles are not similar because similarity transformations preserve angle measure. since the two triangles have different angle measures, they are not similar.
c. yes. the two triangles are similar because one triangle is the reflection of the other.

Explanation:

Step1: Check similarity for part a)

Two triangles are similar if their corresponding angles are equal. Here, the two triangles have a common angle (\(75^{\circ}\)) and the vertical - opposite angles (formed by the intersection of two lines) are equal. By the AA (Angle - Angle) similarity criterion, the two triangles are similar.

Step2: Set up proportion for part b)

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

Step3: Solve for \(d\)

First, calculate \(150\times20=3000\). Then, from \(100d = 3000\), divide both sides by 100: \(d=\frac{3000}{100}=30\).

Answer:

a) Yes. The two triangles are similar because they have two pairs of equal angles (AA similarity criterion).
b) \(30\) ft.