Question

is a triangle with side lengths 106, 56, and 90 a right triangle?

Explanation:

Step1: Recall Pythagorean theorem

For a right triangle, the sum of the squares of the two shorter sides (legs) should equal the square of the longest side (hypotenuse). The formula is \(a^2 + b^2 = c^2\), where \(c\) is the longest side, and \(a\), \(b\) are the shorter sides.
Here, the sides are 106, 56, 90. The longest side is 106, so \(c = 106\), \(a = 56\), \(b = 90\).

Step2: Calculate \(a^2 + b^2\)

First, calculate \(56^2\) and \(90^2\).
\(56^2 = 56\times56 = 3136\)
\(90^2 = 90\times90 = 8100\)
Then, sum them: \(3136 + 8100 = 11236\)

Step3: Calculate \(c^2\)

Calculate \(106^2\): \(106\times106 = 11236\)

Step4: Compare \(a^2 + b^2\) and \(c^2\)

We found that \(a^2 + b^2 = 11236\) and \(c^2 = 11236\). Since \(a^2 + b^2 = c^2\), the triangle satisfies the Pythagorean theorem.

Answer:

Yes, the triangle with side lengths 106, 56, and 90 is a right triangle.