2. Lesson 7-2 Lesson 7-2: The Pythagorean Theorem1 The Pythagorean Theorem

3. Anatomy of a right triangle The hypotenuse of a right triangle is the longest side. It is opposite the right angle. The other two sides are legs. They form the right angle. Lesson 7-2: The Pythagorean Theorem2 hypotenuse leg

4. The Pythagorean Theorem Lesson 7-2: The Pythagorean Theorem 3 a b c 1.Draw a right triangle with lengths a, b and c. (c the hypotenuse) 2. Draw a square on each side of the triangle. 3. What is the area of each square? a2a2 b2b2 c2c2

5. The Pythagorean Theorem Lesson 7-2: The Pythagorean Theorem 4 a b c The Pythagorean Theorem says a 2 + b 2 = c 2 a2a2 b2b2 c2c2

6. Proofs of the Pythagorean Theorem Lesson 7-2: The Pythagorean Theorem5 Proof 1 Proof 2 Proof 3

7. The Pythagorean Theorem Lesson 7-2: The Pythagorean Theorem6 a b c If a triangle is a right triangle, with leg lengths a and b and hypotenuse c, then a 2 + b 2 = c 2 c is the length of the hypotenuse!

8. The Pythagorean Theorem Lesson 7-2: The Pythagorean Theorem7 leg hyp If a triangle is a right triangle, then leg 2 + leg 2 = hyp 2

9. In the following figure if a = 3 and b = 4, Find c. leg 2 + leg 2 = hyp 2 3 2 + 4 2 = C 2 9 + 16 = C 2 25 = C 2 5 = C Example Lesson 7-2: The Pythagorean Theorem8 a b c

10. Pythagorean Theorem : Examples 1.a = 8, b = 15, Find c 2.a = 7, b = 24, Find c 3.a = 9, b = 40, Find c 4.a = 10, b = 24, Find c 5. a = 6, b = 8, Find c Lesson 7-2: The Pythagorean Theorem9 a b c c = 17 c = 25 c = 41 c = 26 c = 10

11. In the following figure if b = 5 and c = 13, Find a. leg 2 + leg 2 = hyp 2 a 2 +5 2 = 13 2 a 2 + 25 = 169 -25 -25 a 2 = 144 a = 12 Finding the legs of a right triangle: Lesson 7-2: The Pythagorean Theorem10 a b c

12. More Examples: 1) a=8, c =10, Find b 2) a=15, c=17, Find b 3) b =10, c=26, Find a 4) a=15, b=20, Find c 5) a =12, c=16, Find b 6) b =5, c=10, Find a 7) a =6, b =8, Find c 8) a=11, c=21, Find b Lesson 7-2: The Pythagorean Theorem11 a b c b = 6 b = 8 a = 24 c = 25 a = 8.7 c = 10

13. A Little More Triangle Anatomy The altitude of a triangle is a segment from a vertex of the triangle perpendicular to the opposite side. Lesson 7-2: The Pythagorean Theorem12 altitude

14. Lesson 3-1: Triangle Fundamentals13 Altitude - Special Segment of Triangle Definition: a segment from a vertex of a triangle perpendicular to the segment that contains the opposite side. In a right triangle, two of the altitudes are the legs of the triangle. B A D F In an obtuse triangle, two of the altitudes are outside of the triangle. B A D F I K

15. Example: An altitude is drawn to the side of an equilateral triangle with side lengths 10 inches. What is the length of the altitude? Lesson 7-2: The Pythagorean Theorem14 h 2 + 5 2 = 10 2 h 2 = 75 h = 10 in h ?5 in

16. The Pythagorean Theorem – in Review Lesson 7-2: The Pythagorean Theorem15 a b c Pythagorean Theorem: If a triangle is a right triangle, with side lengths a, b and c (c the hypotenuse,) then a 2 + b 2 = c 2 What is the converse?

17. The Converse of the Pythagorean Theorem Lesson 7-2: The Pythagorean Theorem16 a b c If, a 2 + b 2 = c 2, then the triangle is a right triangle. C is the LONGEST side!

18. Given a = 6, b=8, and c=10, describe the triangle. Compare a 2 + b 2 and c 2: Since 100 = 100, this is a right triangle. a 2 + b 2 c 2 6 2 + 8 2 10 2 36+ 64 100 100 = Given the lengths of three sides, how do you know if you have a right triangle? Lesson 7-2: The Pythagorean Theorem17 a b c = = =

19. The Contrapositive of the Pythagorean Theorem Lesson 7-2: The Pythagorean Theorem18 a b c If a 2 + b 2  c 2 then the triangle is NOT a right triangle. What if a 2 + b 2  c 2 ?

20. The Contrapositive of the Pythagorean Theorem Lesson 7-2: The Pythagorean Theorem19 a b c If a 2 + b 2  c 2 then either, a 2 + b 2 > c 2 or a 2 + b 2 < c 2 What if a 2 + b 2  c 2 ?

21. The Converse of the Pythagorean Theorem Lesson 7-2: The Pythagorean Theorem20 a b c If a 2 + b 2 > c 2, then the triangle is acute. The longest side is too short!

22. The Converse of the Pythagorean Theorem Lesson 7-2: The Pythagorean Theorem21 ab c If a 2 + b 2 < c 2, then the triangle is obtuse. The longest side is too long!

23. Given a = 4, b = 5, and c =6, describe the triangle. Compare a 2 + b 2 and c 2: Since 41 > 36, this is an acute triangle. a 2 + b 2 c 2 4 2 + 5 2 6 2 16 + 25 36 41 36 > Given the lengths of three sides, how do you know if you have a right triangle? Lesson 7-2: The Pythagorean Theorem22 a b c > > >

24. Given a = 4, b = 6, and c = 8, describe the triangle. Compare a 2 + b 2 and c 2: Since 52 < 64, this is an obtuse triangle. a 2 + b 2 c 2 4 2 + 6 2 8 2 16 + 36 64 52 64 < Given the lengths of three sides, how do you know if you have a right triangle? Lesson 7-2: The Pythagorean Theorem23 a b c < < <

25. Describe the following triangles as acute, right, or obtuse 1) 9, 40, 41 2) 15, 20, 10 3) 2, 5, 6 4) 12,16, 20 5) 14,12,11 6) 2, 4, 3 7) 1, 7, 7 8) 90,150, 120 Lesson 7-2: The Pythagorean Theorem24 a b c right acute obtuse right obtuse acute

26. Application The Distance Formula

27. The Pythagorean Theorem For a right triangle with legs of length a and b and hypotenuse of length c, or

28. The x-axis Start with a horizontal number line which we will call the x- axis. We know how to measure the distance between two points on a number line. x Take the absolute value of the difference: │a – b │ │ – 4 – 9 │= │ – 13 │ = 13

29. The y-axis Add a vertical number line which we will call the y-axis. Note that we can measure the distance between two points on this number line also. x y

30. The Coordinate Plane x y We call the x-axis together with the y-axis the coordinate plane.

31. Coordinates / Ordered Pair Coordinates – numbers that identify the position of a point Ordered Pair – a pair of numbers (x- coordinate, y-coordinate) identifying a point’s position Identify some coordinates and ordered pairs in the diagram. Diagram is from the website www.ezgeometry.com.www.ezgeometry.com

32. Finding Distance in The Coordinate Plane x y We can find the distance between any two points in the coordinate plane by using the Ruler Postulate AND the Pythagorean Theorem. ?

33. Finding Distance in The Coordinate Plane cont. x y First, draw a right triangle. ?

34. x y Next, find the lengths of the two legs. First, the horizontal leg: ? – 48 │(– 4) – 8│= │– 12│ = 12 12

35. Finding Distance in The Coordinate Plane cont. x y So the horizontal leg is 12 units long. Now find the length of the vertical leg: ? 3 – 2 │3 – (– 2)│= │ 5 │ = 5 12 5

36. Finding Distance in The Coordinate Plane cont. x y ? 12 5 Here is what we know so far. Since this is a right triangle, we use the Pythagorean Theorem. 13 The distance is 13 units.

37. The Distance Formula Instead of drawing a right triangle and using the Pythagorean Theorem, we can use the following formula: distance = where (x 1, y 1 ) and (x 2, y 2 ) are the ordered pairs corresponding to the two points. So let’s go back to the example.

38. Example x y Find the distance between these two points. Solution: First : Find the coordinates of each point. ? – 2 – 4 (– 4, – 2) 8 3 (8, 3)

39. Example x y Find the distance between these two points. Solution: First: Find the coordinates of each point. (x 1, y 1 ) = (-4, -2) (x 2, y 2 ) = (8, 3) ? (– 4, – 2) (8, 3)

40. Solution cont. Then: Since the ordered pairs are (x 1, y 1 ) = (-4, -2) and (x 2, y 2 ) = (8, 3) Plug in x 1 = -4, y 1 = -2, x 2 = 8 and y 2 = 3 into distance= = = = 13 Example cont.