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To determine if triangle ABC is a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the side lengths given are:
- Side C: 170 feet
- Side B: 150 feet
- Side A: 80 feet
To check if triangle ABC is a right triangle, we need to compare the sum of the squares of the two shorter sides (sides A and B) to the square of the longest side (side C).
1. Calculate the squares of the side lengths:
- Side A squared: 80^2 = 6400 square feet
- Side B squared: 150^2 = 22500 square feet
2. Add the squares of the two shorter sides:
6400 + 22500 = 28900 square feet
3. Calculate the square of the longest side (side C):
Side C squared: 170^2 = 28900 square feet
Comparing the sum of the squares of the two shorter sides (28900 square feet) to the square of the longest side (28900 square feet), we can conclude that triangle ABC is indeed a right triangle. This is because the sum of the squares of the two shorter sides is equal to the square of the longest side.
Therefore, triangle ABC is a right triangle.