Question

The front of the stage, side C, is 50 feet long. A 40-foot rope runs along the side of square B. A 30-foot rope runs along the side of square A. Is the roped off area, triangle ABC, a right triangle? Explain. (2 points)

e) The diagonal of square A, marked off by the red stars, is where the concession stand is located. A local high school band is performing at the outdoor theater on a summer evening. The band has a school banner that is 40-feet long, and band members would like to hang it across the concession stand to let people know they are performing. Estimate the length of the concession stand to determine if the school banner can fit across the length of the concession stand. Show your work and explain your reasoning. (2 points)

Answer 1

ΔABC is a right triangle

length of the concession stand = 42.4 ft (nearest tenth)

Explanation:

Given:

• Side length of square A = 30 ft
• Side length of square B = 40 ft
• Side length of square C = 50 ft

We can use Pythagoras' Theorem to prove if ΔABC is a right triangle.

Pythagoras' Theorem: a² + b² = c² (where a and b are the legs, and c is the hypotenuse of a right triangle)

Given:

• a = side length of square A = 30
• b = side length of square B = 40
• c = side length of square C = 50

⇒ a² + b² = c²

⇒ 30² + 40² = 50²

⇒ 900 + 1600 = 2500

⇒ 2500 = 2500

Therefore, ΔABC is a right triangle

To find the diagonal of square A, use Pythagoras' Theorem:

Given:

• a = side length of square A = 30
• b = side length of square A = 30
• c = diagonal of square A

⇒ a² + b² = c²

⇒ 30² + 30² = c²

⇒ 900 + 900 = c²

⇒ c² = 1800

⇒ c = √(1800)

⇒ c = 42.42640687

Therefore, the length of the concession stand (diagonal of square A) is 42.4 ft (nearest tenth)