Question

Explain how you can use the converse of the Pythagorean theorem to tell whether three given lengths can be sides of a right triangle.

Short answer

Use the converse of the Pythagorean Theorem which states that for any triangle, if the square of one side is equal to the sum of squares of the other two sides, then the triangle is a right triangle. Identify the longest side, square them all and check the equality \(c^2 = a^2 + b^2\). If true, they can form a right triangle, if not, they can't.

Step-by-step solution

Determine the longest side

Given three lengths, identify the longest side. This potential side will act as the hypotenuse in the Pythagorean theorem.

Apply the Converse of Pythagorean Theorem

Next, square the lengths of all three sides. Using the converse of Pythagorean theorem, check if the square of the longest side is equal to the sum of the squares of the other two sides. Mathematically expressed, this is \(c^2 = a^2 + b^2\), where c is the length of the longest side and a and b are the lengths of the other sides.

Draw a conclusion

If the above equation holds true, then the three lengths can form a right triangle. Otherwise, they cannot.