Question

explain why the triangles are similar. then find the distance represented by x.
why are the triangles similar? choose the correct answer below.
a. there is a pair of congruent vertical angles and the sides that include the two vertical angles are proportional, so the triangles are similar by the sas ~ theorem.
b. there is a pair of congruent vertical angles and a pair of congruent right angles, so the triangles are similar by the aa ~ postulate.
c. there is a pair of congruent vertical angles and the sides that include the two vertical angles are also congruent, so the triangles are similar by the sas ~ theorem.
d. the corresponding sides of two triangles are proportional, so the triangles are similar by the sss ~ theorem.
e. there is a pair of congruent vertical angles and the sides that include the two vertical angles are proportional, so the triangles are similar by the aa ~ postulate.
x = \square feet (type an integer or a simplified fraction.)

Explanation:

Step1: Identify Similar Triangles Ratio

Since the triangles are similar by AA (two angles congruent: vertical angles and right angles), the corresponding sides are proportional. The ratio of the shorter legs to the longer legs should be equal. So, $\frac{100}{x} = \frac{50}{75}$.

Step2: Solve for x

Cross - multiply: $50x = 100\times75$.
Simplify the right - hand side: $50x = 7500$.
Divide both sides by 50: $x=\frac{7500}{50}=150$.

Answer:

150