If triangle DEF has sides measuring 50, 40, and 32, is the triangle a right triangle?

1 answer

To determine whether triangle DEF is a right triangle given its side lengths of 50, 40, and 32, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the longest side (the hypotenuse) is equal to the sum of the squares of the other two sides.

  1. Identify the longest side. In this case, the longest side is 50.

  2. We apply the Pythagorean theorem: \[ a^2 + b^2 = c^2 \] where \(c\) is the longest side (50), and \(a\) and \(b\) are the other two sides (40 and 32).

  3. Calculate the squares of the sides:

    • \(50^2 = 2500\)
    • \(40^2 = 1600\)
    • \(32^2 = 1024\)
  4. Now, check if the sum of the squares of the two shorter sides equals the square of the longest side: \[ 40^2 + 32^2 = 1600 + 1024 = 2624 \]

  5. Compare this sum to \(50^2\): \[ 50^2 = 2500 \]

Since \(2624 \neq 2500\), the triangle is not a right triangle.