Question

Three cities form a triangle. Tom measures the distances between the three cities on a map. The distances between the three cities are 45 miles, 56 miles, and 72 miles.



Is the triangle formed by the three cities a right triangle?

Answer 1

To determine whether a triangle with sides of lengths 45 miles, 56 miles, and 72 miles is a right triangle, we can use the Pythagorean theorem. According to this theorem, a triangle is a right triangle if the square of the length of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides.

First, we identify the sides:
- Let \( a = 45 \) miles
- Let \( b = 56 \) miles
- Let \( c = 72 \) miles (the longest side)

Now, we calculate the squares of the side lengths:
- \( a^2 = 45^2 = 2025 \)
- \( b^2 = 56^2 = 3136 \)
- \( c^2 = 72^2 = 5184 \)

Now, we check if \( c^2 = a^2 + b^2 \):
\[
c^2 = 5184
\]
\[
a^2 + b^2 = 2025 + 3136 = 5161
\]

Since \( 5184 \neq 5161 \), the relationship \( c^2 = a^2 + b^2 \) does not hold true.

Thus, the triangle formed by the three cities is **not a right triangle**.