To determine if triangle ABC with sides measuring 39, 15, and 36 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 longest side) is equal to the sum of the squares of the lengths of the other two sides.
The sides are:
- a = 15
- b = 36
- c = 39 (the longest side, which would be the hypotenuse if the triangle is a right triangle)
Now we apply the Pythagorean theorem: \[ c^2 = a^2 + b^2 \] Calculating each value:
- \(c^2 = 39^2 = 1521\)
- \(a^2 = 15^2 = 225\)
- \(b^2 = 36^2 = 1296\)
Now we sum the squares of the two shorter sides: \[ a^2 + b^2 = 225 + 1296 = 1521 \]
Now check if \(c^2 = a^2 + b^2\): \[ 39^2 = 15^2 + 36^2 \Rightarrow 1521 = 1521 \]
Since both sides are equal, triangle ABC is a right triangle.
Thus, the appropriate response is: Yes, because 1,521 equals 1,521.