Problem 12 - Algebra 1: Concepts and Skills

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.

Step-by-Step Solution

Answer

To check whether three given lengths can be sides of a right triangle, identify the longest side and apply the converse of the Pythagorean theorem. Square the lengths of all sides, and if the square of the longest side equals the sum of the squares of the other two sides, the lengths can form a right triangle.

Step 1: Understand the Pythagorean theorem and its converse

The Pythagorean theorem is a basic rule in geometry used to calculate the length of a side in a right triangle. The theorem states that the square of the length of the hypotenuse \(c\) is equal to the sum of the squares of the other two sides \(a\) and \(b\), expressed as \(c^2 = a^2 + b^2\). The converse of the Pythagorean theorem asserts that if this equation holds true, then the triangle with sides of lengths \(a\), \(b\), and \(c\) is a right triangle.

Step 2: Use the converse of the Pythagorean theorem

Given three lengths, to check if they can compose a right triangle, first identify the longest side. Then, square the lengths of all sides. If the square of the longest side equals the sum of the squares of the other two sides, then the three lengths can form a right triangle according to the converse of the Pythagorean theorem.

Step 3: Examples

For example, for three lengths 3, 4, and 5: The square of 5 is \(5^2 = 25\), and the sum of the squares of 3 and 4 is \(3^2 + 4^2 = 9 + 16 = 25\). Therefore, 3, 4, and 5 can form a right triangle. If the three lengths were 3, 4, and 6, the square of 6 is \(6^2 = 36\), and the sum of the squares of 3 and 4 is \(3^2 + 4^2 = 9 + 16 = 25\). The square of 6 does not equal the sum of the squares of the other two lengths, so 3, 4, and 6 cannot form a right triangle.