1. Example 1 1.What is the area of a square with a side length of 4 inches? x inches? 2.What is the side length of a square with an area of 25 in 2 ? x in 2 ? x x

2. Parts of a Right Triangle Which segment is the longest in any right triangle?

3. Apply the Pythagorean Theorem Objectives: 1.To discover and use the Pythagorean Theorem 2.To use Pythagorean Triples to find quickly find a missing side length in a right triangle

4. The Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

5. The Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

6. The Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

7. Example 2 The triangle below is definitely not a right triangle. Does the Pythagorean Theorem work on it?

8. Example 3 How high up on the wall will a twenty- foot ladder reach if the foot of the ladder is placed five feet from the wall?

9. Example 4: SAT In figure shown, what is the length of RS?

10. Example 5 What is the area of the large square?

11. Example 6 Find the area of the triangle.

12. Pythagorean Triples Pythagorean Triples Three whole numbers that work in the Pythagorean formulas are called Pythagorean Triples.

13. Example 7 What happens if you add the same length to each side of a right triangle? Do you still get another right triangle?

14. Example 8 What happens if you multiply all the side lengths of a right triangle by the same number? Do you get another right triangle?

15. Pythagorean Multiples Pythagorean Multiples Conjecture: If you multiply the lengths of all three sides of any right triangle by the same number, then the resulting triangle is a right triangle. In other words, if a 2 + b 2 = c 2, then (an) 2 + (bn) 2 = (cn) 2.

16. Pythagorean Triples Pythagorean Triples

17. Primitive Pythagorean Triples primitive Pythagorean triple A set of Pythagorean triples is considered a primitive Pythagorean triple if the numbers are relatively prime; that is, if they have no common factors other than 1. 3-4-55-12-137-24-258-15-17 9-40-4111-60-6112-35-3713-84-85 16-63-6520-21-2928-45-5333-56-65 36-77-8539-80-8948-55-7365-72-97

18. Example 9 Find the length of one leg of a right triangle with a hypotenuse of 35 cm and a leg of 28 cm.

19. Example 10 Use Pythagorean Triples to find each missing side length.

20. Example 11 A 25-foot ladder is placed against a building. The bottom of the ladder is 7 feet from the building. If the top of the ladder slips down 4 feet, how many feet will the bottom slip out?

21. Converse of the Pythagorean Theorem Objectives: 1.To investigate and use the Converse of the Pythagorean Theorem 2.To classify triangles when the Pythagorean formula is not satisfied

22. Theorem! Converse of the Pythagorean Theorem If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then it is a right triangle.

23. Example 3 Which of the following is a right triangle?

24. Example 4 Tell whether a triangle with the given side lengths is a right triangle. 1.5, 6, 7 2.5, 6, 3.5, 6, 8

25. Theorems! Acute Triangle Theorem 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 it is an acute triangle.

26. Theorems! Obtuse Triangle Theorem 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 it is an obtuse triangle.

27. Example 5 Can segments with lengths 4.3 feet, 5.2 feet, and 6.1 feet form a triangle? If so, would the triangle be acute, right, or obtuse?

28. Example 7 The sides of an obtuse triangle have lengths x, x + 3, and 15. What are the possible values of x if 15 is the longest side of the triangle?

29. Special Right Triangles Objectives: 1.To use the properties of 45-45-90 and 30-60-90 right triangles to solve problems

30. Investigation 1 This triangle is also referred to as a 45-45-90 right triangle because each of its acute angles measures 45°. Folding a square in half can make one of these triangles. In this investigation, you will discover a relationship between the lengths of the legs and the hypotenuse of an isosceles right triangle.

31. Investigation 1 Find the length of the hypotenuse of each isosceles right triangle. Simplify the square root each time to reveal a pattern.

32. Investigation 1 Did you notice something interesting about the relationship between the length of the hypotenuse and the length of the legs in each problem of this investigation?

33. Special Right Triangle Theorem 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, the hypotenuse is times as long as each leg.

34. Example 1 Use deductive reasoning to verify the Isosceles Right Triangle Conjecture.

35. Example 2 A fence around a square garden has a perimeter of 48 feet. Find the approximate length of the diagonal of this square garden.

36. Investigation 2 The second special right triangle is the 30- 60-90 right triangle, which is half of an equilateral triangle. Let’s start by using a little deductive reasoning to reveal a useful relationship in 30-60- 90 right triangles.

37. Investigation 3 Triangle ABC is equilateral, and segment CD is an altitude. 1.What are m<A and m<B? 2.What are m<ADC and m<BDC? 3.What are m<ACD and m<BCD? 4.Is Δ ADC = Δ BDC? Why? 5.Is AD=BD? Why? ~

38. Investigation 2 Notice that altitude CD divides the equilateral triangle into two right triangles with acute angles that measure 30° and 60°. Look at just one of the 30-60-90 right triangles. How do AC and AD compare?Conjecture: In a 30°-60°-90° right triangle, if the side opposite the 30° angle has length x, then the hypotenuse has length -?-.

39. Investigation 2 Find the length of the indicated side in each right triangle by using the conjecture you just made.

40. Investigation 2 Now use the previous conjecture and the Pythagorean formula to find the length of each indicated side.

41. Investigation 2 You should have notice a pattern in your answers. Combine your observations with you latest conjecture and state your next conjecture.

42. Special Right Triangle Theorem 30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg.

43. Two Special Right Triangles

44. Example 3 Find the value of each variable. Write your answer in simplest radical form. 1. 2. 3.

45. Example 4 Find the value of each variable. Write your answer in simplest radical form. 1. 2. 3.

46. Example 5 What is the area of an equilateral triangle with a side length of 4 cm? 4 cm

47. Example 6: SAT In the figure, what is the ratio of RW to WS?