Question

a building casts a shadow 40 feet long, and a nearby 8-foot pole casts a shadow 2 feet long. what is the height of the building?

a. 180 feet
b. 160 feet
c. 150 feet
d. 200 feet

what is the sum of the interior angles in a triangle?

a. 360 degrees
b. 180 degrees
c. 270 degrees
d. 90 degrees

a triangle with a base of 5 feet and a height of 8 feet is similar to a triangle with a base of 15 feet. what is the height of the larger triangle?

a. 24 feet
b. 32 feet
c. 20 feet
d. 16 feet

Explanation:

First Question (Building and Pole Shadow)

Step1: Set up proportion

Let \( h \) be the height of the building. The ratio of height to shadow length should be equal for the building and the pole. So, \(\frac{h}{40}=\frac{8}{2}\)

Step2: Solve for \( h \)

Cross - multiply: \( 2h = 40\times8 \)
\( 2h=320 \)
Divide both sides by 2: \( h=\frac{320}{2}=160 \)

By the triangle angle - sum theorem, the sum of the interior angles of any triangle is always 180 degrees. This is a fundamental geometric theorem.

Step1: Set up proportion for similar triangles

For similar triangles, the ratios of corresponding sides are equal. Let \( H \) be the height of the larger triangle. So, \(\frac{8}{5}=\frac{H}{15}\)

Step2: Solve for \( H \)

Cross - multiply: \( 5H = 8\times15 \)
\( 5H = 120 \)
Divide both sides by 5: \( H=\frac{120}{5}=24 \)

Answer:

b. 160 feet

Second Question (Sum of Interior Angles of a Triangle)