Lesson 8.3 Pythagorean Theorem pp. 323-327

Lesson 8.3Pythagorean Theorem

pp. 323-327

Lesson 8.3Pythagorean Theorem

pp. 323-327

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.

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.

Theorem 8.7Pythagorean 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.

Theorem 8.7Pythagorean 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.

The Pythagorean Theorem can also be thought of as

leg2 + leg2 = hypotenuse2

The Pythagorean Theorem can also be thought of as

leg2 + leg2 = hypotenuse2

The following illustrates the Pythagorean theorem.The following illustrates the Pythagorean theorem.

bb

bb

bb

bb

aa

aa

aaaa

cc

cc

cc

cc

Area of large square: A = (a+b)2

Area of small square: A = c2

Area of triangle: A = ½ab

Area of large square: A = (a+b)2

Area of small square: A = c2

Area of triangle: A = ½ab

bb

bb

bb

bb

aaaa

aaaa

cc

cccc

cc

Area of large square = Area of small square + 4 x Area of triangle.Area of large square = Area of small square + 4 x Area of triangle.

bb

bb

bb

bb

aaaa

aaaa

cc

cccc

cc

bb

bb

bb

bb

aaaa

aaaa

cc

cccc

cc

(a+b)2 = c2 + 4(½ab)

a2 + 2ab + b2 = c2 + 2ab

a2 + b2 = c2

(a+b)2 = c2 + 4(½ab)

a2 + 2ab + b2 = c2 + 2ab

a2 + b2 = c2

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.

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.

Theorem 8.8 Theorem 8.8

The area of an equilateral triangle is

times the square of the length of

one side: A = s2

The area of an equilateral triangle is

times the square of the length of

one side: A = s2

4433

4433

AA DD BB

CC

ss sshh

ss

½s½s

h2 + (½s)2 = s2

h2 + ¼s2 = s2

4h2 + s2 = 4s2

4h2 = 3s2

h2 = ¾s2

h =

h2 + (½s)2 = s2

h2 + ¼s2 = s2

4h2 + s2 = 4s2

4h2 = 3s2

h2 = ¾s2

h = s 3

2s 3

2

AA DD BB

CC

ss sshh

ss

A = ½bhA = ½bhs 3

2s 3

2A =A =1212

(s)(s)

s2 34

s2 34A =A =

EXAMPLE Find the area of equilateral LMN.EXAMPLE Find the area of equilateral LMN.

LL

MMNN

7 cm7 cm

A = s2A = s2 34

34

A = 72A = 72 34

34

A = sq. cm A = sq. cm 49 34

49 34

Practice: A screen door measures 36 in. by 80 in. Find the length of a diagonal brace for the screen door.

Practice: A screen door measures 36 in. by 80 in. Find the length of a diagonal brace for the screen door.

80 in.80 in.

36 in.36 in.

cc

362 + 802 = c2

7,696 = c2

c ≈ 87.7 in.

362 + 802 = c2

7,696 = c2

c ≈ 87.7 in.

Homeworkpp. 325-327Homeworkpp. 325-327

a(units) b(units) c(units)1. 3 53. 9 25. 2 37. 6 49. x 5


►A. ExercisesComplete the table. Consider ABC to be a right triangle with c as the hypotenuse.


44

13132255

8585

x2 + 25x2 + 25





4485 or 9.285 or 9.2

x2 + 25x2 + 25

5 or 2.25 or 2.252 or 7.252 or 7.2

►A. ExercisesTell which of the following triangles are right triangles.11.


9966

88

62 + 82 9262 + 82 92


►A. ExercisesTell which of the following triangles are right triangles.13. 202 + 212 = 292202 + 212 = 292

20202121

2929

►B. ExercisesFind each area.15. A right triangle has a leg

measuring 5 inches and a hypotenuse measuring 8 inches.

►B. ExercisesFind each area.15. A right triangle has a leg

measuring 5 inches and a hypotenuse measuring 8 inches.

5588

►B. Exercises17. Find the area.►B. Exercises17. Find the area. 1515

1212

55

►B. Exercises19. Find the area.►B. Exercises19. Find the area.

5555

55

►B. Exercises21. The bases of an isosceles

trapezoid are 6 inches and 12 inches. Find the area if the congruent sides are 5 inches.

►B. Exercises21. The bases of an isosceles

trapezoid are 6 inches and 12 inches. Find the area if the congruent sides are 5 inches.

6

12

5

►B. Exercises22. The perimeter of a rhombus is 52

and one diagonal is 24. Find the area.



1313

2424

1313

121255





1313

2424

1010

A = ½d1d2

= ½(10)(24)= 120

A = ½d1d2

= ½(10)(24)= 120





■ Cumulative ReviewFind the area of each rectangle.29.

■ Cumulative ReviewFind the area of each rectangle.29. 77

33



7744

■ Cumulative Review31. Give bounds for c if b 5.

■ Cumulative Review31. Give bounds for c if b 5.

77

ccbb

■ Cumulative Review32. The consecutive sides of a rectangle

have a ratio 4:5. If the area is 5120 m2, what are the dimensions of the

rectangle?

■ Cumulative Review32. The consecutive sides of a rectangle

have a ratio 4:5. If the area is 5120 m2, what are the dimensions of the

rectangle?

■ Cumulative Review33. The figure is made up of 8 congruent

squares and has a total area of 968 cm2. Find the perimeter.

■ Cumulative Review33. The figure is made up of 8 congruent

squares and has a total area of 968 cm2. Find the perimeter.

Analytic Geometry

Heron’s Formula

Analytic Geometry

Heron’s Formula

ss ..22

ccbbaa ++++==

The semiperimeter of a triangle is one-half the perimeter of a triangle:The semiperimeter of a triangle is one-half the perimeter of a triangle:

DefinitionDefinitionDefinitionDefinition

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) .

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) .

Find the area of a triangle with sides 9, 10, and 15.Find the area of a triangle with sides 9, 10, and 15.

1717==22

1515101099ss

++++==

= 17(17 – 9)(17 – 10)(17 – 15)= 17(17 – 9)(17 – 10)(17 – 15)

Find the area of a triangle with sides 9, 10, and 15.Find the area of a triangle with sides 9, 10, and 15.

≈43.6 sq. units≈43.6 sq. units

= 1904= 1904

= 17(8)(7)(2)= 17(8)(7)(2)

A = s(s – a)(s – b)(s – c)A = s(s – a)(s – b)(s – c)

►ExercisesFind 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


Lesson 8.3 Pythagorean Theorem pp. 323-327

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