1. Lesson 8.3
Pythagorean Theorem pp

2. Objectives: 1. To prove the Pythagorean theorem.
2. To apply the Pythagorean theorem to right triangles.
3. To develop a formula for the area of an equilateral triangle.

3. Theorem 8.7
Pythagorean Theorem. In a right triangle, the sum of the squares of the length of the legs is equal to the square of the length of the hypotenuse: a2 + b2 = c2.

4. The Pythagorean Theorem can also be thought of as
leg2 + leg2 = hypotenuse2

5. The following illustrates the Pythagorean theorem.

6. b a c

7. b a c Area of large square: A = (a+b)2 Area of small square: A = c2
Area of triangle: A = ½ab

8. b a c
Area of large square = Area of small square + 4 x Area of triangle.

9. b a c
(a+b)2 = c2 + 4(½ab)
a2 + 2ab + b2 = c2 + 2ab
a2 + b2 = c2

10. The converse of the Pythagorean theorem is also true
The converse of the Pythagorean theorem is also true. If the sum of the squares of the lengths of two sides of a triangle equals the square of the length of the third side, then the triangle is a right triangle.

11. Theorem 8.8 The area of an equilateral triangle is
times the square of the length of one side: A = s2 4 3

12. A D B C s h ½s
h2 + (½s)2 = s2
h2 + ¼s2 = s2
4h2 + s2 = 4s2
4h2 = 3s2
h2 = ¾s2
h =
s 3 2

13. A D B C s h
A = ½bh
s 3 2
A = 1
(s)
s2 3 4
A =

14. EXAMPLE Find the area of equilateral LMN.
A = s2 3 4 L M N
7 cm
A = 72 3 4
A = sq. cm
49 3 4

15. Practice: A screen door measures 36 in. by 80 in
Practice: A screen door measures 36 in. by 80 in. Find the length of a diagonal brace for the screen door.
80 in.
36 in. c
= c2
7,696 = c2
c ≈ 87.7 in.

16. Homework pp

17. 85 5 13 2 x2 + 25 ►A. Exercises a(units) b(units) c(units) 1. 3 5
Complete the table. Consider ABC to be a right triangle with c as the hypotenuse.
a(units) b(units) c(units)
3. 9 2
5. 2 3
7. 6 4
9. x 5 4 85 5 13 2
x2 + 25

18. x2 + 25 ►A. Exercises a(units) b(units) c(units) 1. 3 5 3. 9 2 5. 2 3
Complete the table. Consider ABC to be a right triangle with c as the hypotenuse.
a(units) b(units) c(units)
3. 9 2
5. 2 3
7. 6 4
9. x 5 4
85 or 9.2
5 or 2.2
52 or 7.2
x2 + 25

19. Tell which of the following triangles are right triangles. 11.
►A. Exercises
Tell which of the following triangles are right triangles.
11.
 92 9 6 8

20. Tell which of the following triangles are right triangles. 13.
►A. Exercises
Tell which of the following triangles are right triangles.
13.
= 292 20 21 29

21. ►B. Exercises
Find each area.
15. A right triangle has a leg measuring 5 inches and a hypotenuse measuring 8 inches. 5 8

22. ►B. Exercises
17. Find the area. 15 12 5

23. ►B. Exercises
19. Find the area. 5

24. ►B. Exercises
21. The bases of an isosceles trapezoid are 6 inches and inches. Find the area if the congruent sides are 5 inches. 6 12 5

25. ►B. Exercises
22. The perimeter of a rhombus is 52 and one diagonal is 24. Find the area. 24 13

26. ►B. Exercises
22. The perimeter of a rhombus is 52 and one diagonal is 24. Find the area. 12 5 13

27. ►B. Exercises
22. The perimeter of a rhombus is 52 and one diagonal is 24. Find the area.
A = ½d1d2
= ½(10)(24)
= 120 24 10 13

28. ■ Cumulative Review
Find the area of each rectangle.
29. 7 3

29. ■ Cumulative Review
Find the area of each rectangle.
30. 7 4

30. ■ Cumulative Review
31. Give bounds for c if b  5. c b 7

31. ■ Cumulative Review
32. The consecutive sides of a rectangle have a ratio 4:5. If the area is 5120 m2, what are the dimensions of the rectangle?

32. ■ Cumulative Review
33. The figure is made up of 8 congruent squares and has a total area of cm2. Find the perimeter.

33. Analytic Geometry
Heron’s Formula

34. Definition
The semiperimeter of a triangle is one-half the perimeter of a triangle: s . 2 c b a + =

35. Heron’s Formula
If ABC has sides of lengths a, b, and c and semiperimeter s, then the area of the triangle is
A = s(s - a)(s - b)(s - c) .

36. Find the area of a triangle with sides 9, 10, and 15.2 15 10 9 s + = 17 =

37. Find the area of a triangle with sides 9, 10, and 15.
A = s(s – a)(s – b)(s – c)
= 17(17 – 9)(17 – 10)(17 – 15)
= 17(8)(7)(2) =
≈ 43.6 sq. units

38. ►Exercises 1. 3 units, 8 units, 9 units
Find the area of each triangle that has the given side measures. Round you answers to the nearest tenth.
1. 3 units, 8 units, 9 units

39. ►Exercises 2. 6 units, 18 units, 21 units
Find the area of each triangle that has the given side measures. Round you answers to the nearest tenth.
2. 6 units, 18 units, 21 units

40. ►Exercises 3. 27 units, 13 units, 18 units
Find the area of each triangle that has the given side measures. Round you answers to the nearest tenth.
3. 27 units, 13 units, 18 units