Objectives/Assignment

1. Objectives/Assignment • Prove the Pythagorean Theorem • Use the Pythagorean Theorem to solve real-life problems such as determining how far a ladder will reach. • Assignment: pp. 538-539 #1-31 all • Assignment due today: 9.1 pp. 531-532 #1-34 all

2. History Lesson • Around the 6th century BC, the Greek mathematician Pythagorus founded a school for the study of philosophy, mathematics and science. Many people believe that an early proof of the Pythagorean Theorem came from this school. • Today, the Pythagorean Theorem is one of the most famous theorems in geometry. Over 100 different proofs now exist.

3. Proving the Pythagorean Theorem • In this lesson, you will study one of the most famous theorems in mathematics—the Pythagorean Theorem. The relationship it describes has been known for thousands of years.

4. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the legs. Theorem 9.4: Pythagorean Theorem c2 = a2 + b2

5. Given: In ∆ABC, BCA is a right angle. Prove: c2 = a2 + b2 Proof: Plan for proof: Draw altitude CD to the hypotenuse just like in 9.1. Then apply Geometric Mean Theorem 9.3 which states that when the altitude is drawn to the hypotenuse of a right triangle, each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

6. Statements: 1. Draw a perpendicular from C to AB. 2. and Reasons: Proof • Perpendicular Postulate • Geometric Mean Thm. • Cross Product Property c a c b = = a e b f 3. ce = a2 and cf = b2 4. ce + cf = a2 + b2 4. Addition Property of = 5. c(e + f) = a2 + b2 5. Distributive Property 6. Segment Add. Postulate 6. e + f = c 7. c2 = a2 + b2 7. Substitution Property of =

7. Using the Pythagorean Theorem • A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation c2 = a2 + b2For example, the integers 3, 4 and 5 form a Pythagorean Triple because 52 = 32 + 42.

8. Find the length of the hypotenuse of the right triangle. Tell whether the sides lengths form a Pythagorean Triple. Ex. 1: Finding the length of the hypotenuse.

9. (hypotenuse)2 = (leg)2 + (leg)2 x2 = 52 + 122 x2 = 25 + 144 x2 = 169 x = 13 Because the side lengths 5, 12 and 13 are integers, they form a Pythagorean Triple. Many right triangles have side lengths that do not form a Pythagorean Triple as shown next slide. Pythagorean Theorem Substitute values. Multiply Add Find the positive square root. Note: There are no negative square roots until you get to Algebra II and introduced to “imaginary numbers.” Solution:

10. Find the length of the leg of the right triangle. Ex. 2: Finding the Length of a Leg

11. (hypotenuse)2 = (leg)2 + (leg)2 142 = 72 + x2 196 = 49 + x2 147 = x2 √147 = x √49 ∙ √3 = x 7√3 = x Pythagorean Theorem Substitute values. Multiply Subtract 49 from each side Find the positive square root. Use Product property Simplify the radical. Solution: • In example 2, the side length was written as a radical in the simplest form. In real-life problems, it is often more convenient to use a calculator to write a decimal approximation of the side length. For instance, in Example 2, x = 7 ∙√3 ≈ 12.1

12. Find the area of the triangle to the nearest tenth of a meter. You are given that the base of the triangle is 10 meters, but you do not know the height. Ex. 3: Finding the area of a triangle Because the triangle is isosceles, it can be divided into two congruent triangles with the given dimensions. Use the Pythagorean Theorem to find the value of h.

13. Steps: (hypotenuse)2 = (leg)2 + (leg)2 72 = 52 + h2 49 = 25 + h2 24 = h2 √24 = h Reason: Pythagorean Theorem Substitute values. Multiply Subtract 25 both sides Find the positive square root. Solution: Now find the area of the original triangle.

14. Area of a Triangle Area = ½ bh = ½ (10)(√24) ≈ 24.5 m2 The area of the triangle is about 24.5 m2

15. Support Beam: The skyscrapers shown on page 535 are connected by a skywalk with support beams. You can use the Pythagorean Theorem to find the approximate length of each support beam. Ex. 4: Indirect Measurement

16. Each support beam forms the hypotenuse of a right triangle. The right triangles are congruent, so the support beams are the same length. Use the Pythagorean Theorem to show the length of each support beam (x).

17. (hypotenuse)2 = (leg)2 + (leg)2 x2 = (23.26)2 + (47.57)2 x2 = √ (23.26)2 + (47.57)2 x ≈ 13 Pythagorean Theorem Substitute values. Multiply and find the positive square root. Use a calculator to approximate. Solution:

18. Objectives/Assignment • Use the Converse of the Pythagorean Theorem to solve problems. • Use side lengths to classify triangles by their angle measures. • Assignment: pp. 545-547 #1-35 • Assignment due today: 9.2 • Reminder Quiz after this section on Monday.

19. Using the Converse • In Lesson 9.2, 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.

20. 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. Converse of the Pythagorean Theorem

21. 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.

22. 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

23. 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.

24. 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.

25. 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.

26. 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 Triangle Inequality c2 < a2 + b2

27. 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 Triangle Inequality c2 > a2 + b2

28. 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.

29. 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 confirm

30. 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 confirm

31. 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

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

33. 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.

34. 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