8.1 The Converse of the Pythagorean Theorem

Geometry Mr. Peebles Spring 2013 Bell Ringer You are to walk around the room and locate and write the 10 BEs for math success. Bell Ringer You are to walk around the room and locate and write the 10 BEs for math success.

Time’s Up! Bell Ringer-Answer The 10 Non-Negotiable “BEs” For Geometry Success

1. BE on time and in class every day. 2. BE dressing for success. 3. BE respectful… give it, earn it. 4. BE prepared with your own work and materials. 5. BE participating as a team player. 6. BE meeting all course due dates. 7. BE electronics, makeup, food, drink, candy, and gum-free. 8. BE following all teacher directions. 9. BE following all class, school, and district rules. 10. BE expecting “necessary” changes. Daily Learning Target (DLT) • “I can apply my knowledge of right triangles and Pythagorean Theorem to determine if a triangle is a right triangle.” Pre-Assessment On the piece of paper, write down what you Know about the following topics.

1. Converse of Pythagorean Theorem 2. 45-45-90 Right Triangles 3. 30-60-90 Right Triangles 4. Tangent Ratio 5. Sine Ratio 6. Cosine Ration

Using the Converse

• In the past, you learned that 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, as stated on the following slide. Theorem 9.5: Converse of the Pythagorean Theorem

• If the square of the length B of the longest side of the triangle is equal to the sum of the squares of the lengths of the other two a c sides, then the triangle is a right triangle. C b A • If c2 = a2 + b2, then ∆ABC is a right triangle. Note:

• You can use the Converse of the Pythagorean Theorem to verify that a given triangle is a right triangle, as shown in Example 1. Ex. 1: Verifying Right Triangles

• The triangles on the slides that follow appear to be right 8 7 triangles. Tell whether they are right triangles or not. √113

4√95 15

36 Ex. 1a: Verifying Right Triangles

• Let c represent the length of the longest side of the triangle. 8 Check to see whether 7 the side lengths 2 satisfy the equation c √113 = a2 + b2. (√113)2 =? 72 + 82 113 =? 49 + 64 The triangle is a right triangle. 113 = 113 ✔ Ex. 1b: Verifying Right Triangles

2 2 2 c = a + b . 4√95 ? (4√95)2 = 152 + 362 15 ? 42 ∙ (√95)2 = 152 + 362 ? 36 16 ∙ 95 = 225+1296 1520 ≠ 1521 ✔ 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. Theorem 9.6—Triangle Inequality

• If the square of the A length of the longest side of a triangle is c less than the sum of b the squares of the lengths of the other two sides, then the C a B triangle is acute. 2 2 2 • If c2 < a2 + b2, then c < a + b ∆ABC is acute Theorem 9.7—Triangle Inequality

• If the square of the A length of the longest c side of a triangle is greater than the sum b of the squares of the lengths of the other C B a two sides, then the triangle is obtuse. 2 2 2 • If c2 > a2 + b2, then c > a + b ∆ABC is obtuse 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. a. 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. Triangle Inequality to confirm Example 2a Statement: Reason: c2 ? a2 + b2 Compare c2 with a2 + b2 862 ? 382 + 772 Substitute values 7396 ? 1444 + 5959 Multiply 7395 > 7373 c2 is greater than a2 + b2

The triangle is obtuse Triangle Inequality to confirm Example 2b Statement: Reason: c2 ? a2 + b2 Compare c2 with a2 + b2 37.52 ? 10.52 + 36.52 Substitute values 1406.25 ? 110.25 + Multiply 1332.25 1406.24 < 1442.5 c2 is less than a2 + b2 The triangle is acute Ex. 3: Building a foundation

• 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

• 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. Ex. 3: Building a foundation

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

• Pgs. 420-423 (11-17 Odds, 21-29 All, 31, 54, 59, 60) Closure:

• Given three sides, how would you know if a triangle is acute, right, or obtuse? - Please write your answer on the whiteboards.

Closure:

• Given three sides, how would you know if a triangle is acute, right, or obtuse? - Please write your answer on the whiteboards. c2 < a2 + b2 = Acute Triangle c2 = a2 + b2 = Right Triangle c2 > a2 + b2 = Obtuse Triangle