The Converse of the Pythagorean Theorem

The Converse of the Pythagorean Theorem Geometry Mrs. Spitz Spring 2005

Objectives • Use the Converse of the Pythagorean Theorem to solve problems. • Use side lengths to classify triangles by their angle measures.

Using the Converse • If a triangle is a right triangle, then the square of the length of the hypotenuse is equal to the sum of the squares of the length of the legs. The Converse of the Pythagorean Theorem is also true.

If the square of the length of the longest side of the triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. If c2 = a2 + b2, then ∆ABC is a right triangle. Theorem: Converse of the Pythagorean Theorem

Note: • The Converse of the Pythagorean Theorem can be used to verify that a given triangle is a right triangle.

The triangles on the slides that follow appear to be right triangles. Tell whether they are right triangles or not. Ex. 1: Verifying Right Triangles √113 4√95

Let c represent the length of the longest side of the triangle. Check to see whether the side lengths satisfy the equation c2 = a2 + b2. (√113)2 = 72 + 82 113 = 49 + 64 113 = 113 ✔ Ex. 1a: Verifying Right Triangles √113 ? ? The triangle is a right triangle.

c2 = a2 + b2. (4√95)2 = 152 + 362 42 ∙ (√95)2 = 152 + 362 16 ∙ 95 = 225+1296 1520 ≠ 1521 ✔ Ex. 1b: Verifying Right Triangles 4√95 ? ? ? The triangle is NOT a right triangle.

Classifying Triangles • Sometimes it is hard to tell from looking at a triangle whether it is obtuse or acute. The theorems on the following slides can help you tell.

If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute. If c2 < a2 + b2, then ∆ABC is acute Theorem — Triangle Inequality c2 < a2 + b2

If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse. If c2 > a2 + b2, then ∆ABC is obtuse Theorem — Triangle Inequality c2 > a2 + b2

Ex. 2: Classifying Triangles • Decide whether the set of numbers can represent the side lengths of a triangle. If they can, classify the triangle as right, acute or obtuse. • 38, 77, 86 b. 10.5, 36.5, 37.5 You can use the Triangle Inequality to confirm that each set of numbers can represent the side lengths of a triangle. Compare the square o the length of the longest side with the sum of the squares of the two shorter sides.

Statement: c2? a2 + b2 862? 382 + 772 7396 ? 1444 + 5959 7395 > 7373 Reason: Compare c2 with a2 + b2 Substitute values Multiply c2 is greater than a2 + b2 The triangle is obtuse Triangle Inequality to confirmExample 2a

Statement: c2? a2 + b2 37.52? 10.52 + 36.52 1406.25 ? 110.25 + 1332.25 1406.24 < 1442.5 Reason: Compare c2 with a2 + b2 Substitute values Multiply c2 is less than a2 + b2 The triangle is acute Triangle Inequality to confirmExample 2b

Construction: You use four stakes and string to mark the foundation of a house. You want to make sure the foundation is rectangular. a. A friend measures the four sides to be 30 feet, 30 feet, 72 feet, and 72 feet. He says these measurements prove that the foundation is rectangular. Is he correct? Ex. 3: Building a foundation

Ex. 3: Building a foundation • Solution: Your friend is not correct. The foundation could be a nonrectangular parallelogram, as shown below.

Ex. 3: Building a foundation b. You measure one of the diagonals to be 78 feet. Explain how you can use this measurement to tell whether the foundation will be rectangular.

Solution: The diagonal divides the foundation into two triangles. Compare the square of the length of the longest side with the sum of the squares of the shorter sides of one of these triangles. Because 302 + 722 = 782, you can conclude that both the triangles are right triangles. The foundation is a parallelogram with two right angles, which implies that it is rectangular Ex. 3: Building a foundation

The Converse of the Pythagorean Theorem