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?
40 ft
200 ft
120 ft
d. yes. the two triangles are similar because there is a transformation, composed of rigid motions and dilations, that will take one triangle to the other.
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.

Explanation:

Step1: Verify triangle similarity

The two triangles have a pair of equal vertical angles, and the given angles (40° each) are equal. By AA (Angle-Angle) similarity criterion, the triangles are similar.

Step2: Set up proportion for sides

Corresponding sides of similar triangles are proportional. Let the lake distance be $d$.
$\frac{d}{200} = \frac{130}{40}$

Step3: Solve for $d$

Multiply both sides by 200 to isolate $d$.
$d = 200 \times \frac{130}{40}$
$d = 5 \times 130 = 650$

Answer:

a) Yes. The two triangles are similar by the AA (Angle-Angle) similarity criterion: they share a pair of equal vertical angles, and each has a 40° angle, so all corresponding angles are equal.
b) A. The distance across the lake is 650 ft. (Simplify your answer.)