Geometry 8.2 The Pythagorean Theorem (This section, along with 8.4, are very important as they are...

GeometryGeometry

8.2 The Pythagorean Theorem8.2 The Pythagorean Theorem(This section, along with 8.4, are very important as they are (This section, along with 8.4, are very important as they are

utilized throughout the second semester)utilized throughout the second semester)

Radical ReviewRadical Review

Simplify each Simplify each expression.expression. 2

63

2 )2

2)5( )1

2

5

3 )4

You try!

2)7(2 )3

= 5 = 8/3

= 28 = 9/5

Do you know these?Do you know these?

7 =2

49

8 =2

64

14 =2

196

15 =2

225

12 =2

144 10 =2

100

13 =2

169

9 =2

81

11 =2

121 17 =2

289

16 =2

256

Pythagorean TheoremPythagorean Theorem

In a right triangle, the square of the In a right triangle, the square of the hypotenuse is equal to the sum of the hypotenuse is equal to the sum of the squares of the legs. squares of the legs.

a

bc

C

A

B

222 cba

Pythagorean Theorem ProofPythagorean Theorem ProofThis is in the state standards and may be on the STAR test!It is important to get this down!

b

c

a

aa

a

b

b

b

c

c

c

½ ab

½ ab

½ ab

½ ab(b – a)

(b – a)

(b – a)

(b – a)

Area of large square = Area of large square

c 2 =

4 triangles + small square

c 2 =

4(½ ab) +

(b – a)(b – a)

b – 2ab + a2 2

b – 2ab + a2 2

large square =

(b – 2ab + a )2 2

2ab + b – 2ab + a 2 2

c 2 = b + a 2 2

c c

c

c

c 2=a + b 2 2

Find the value of x together.

C A

B

# BC AC AB

1) 8 6 x

2) x 9 15

7) 12 x

x = 10

x = 12

x = 38 34

Find the value of x on your own.

C A

B

# BC AC AB

3) 5 5 x

4) x 3

Try #3, 4 and 8. Who can solve these on the board?

x =

x =

25

6) 1 x x =

5) x 1 x = 1 2

8) x 8 x =

6

12

2

33

3

54

Solve for x.

x45

3

6

x

9)

6

3 + 6 = x2 2 2

9 + 36 = x2

45 = x2

9 5

3 3

53 x

Solve for x.16)

y

15 + y = 172 2 2

225 + y = 2892

y = 642

15

17

6

x

y = 8

6 + 8 = x2 2 2

You may recognize this one, x = 10.

18) The diagonals of a rhombus have lengths 18 and 24. Find the perimeter of the rhombus.

A rhombus has perpendicular diagonals.

A rhombus is a parallelogram. Diagonals of a parallelogram bisecteach other.

9

12

9 + 12 = x2 2 2

x

81 + 144 = x2

225 = x2

x = 15

1515

15 15Thus, the perimeter is 60.

RemindersReminders

The diagonals of a rhombus are The diagonals of a rhombus are perpendicular to each other.perpendicular to each other.

The altitude drawn to the base of an The altitude drawn to the base of an isosceles triangle is perpendicular to the isosceles triangle is perpendicular to the base at its midpoint.base at its midpoint.

HWHW

Complete #1-20 from the packet on a Complete #1-20 from the packet on a separate sheet. For any ones we have separate sheet. For any ones we have done already, write “Did in Class”done already, write “Did in Class”

P. 292 #19-22 P. 292 #19-22 P. 289 (32-38 Even)P. 289 (32-38 Even)

Part of the HW, Answers to exercises Part of the HW, Answers to exercises #10-20 Skip 14, 16, 18, 20.#10-20 Skip 14, 16, 18, 20.

10) x = 1710) x = 17

11) x = 411) x = 4

12) x = 312) x = 3

13) x = 813) x = 8

14)14)

15)15)

17) 17)

19)19)

20) x = 7 20) x = 7

105x

38x214x

22x

Let’s review the triangle proportion Let’s review the triangle proportion formulas from 8.1 and do # on formulas from 8.1 and do # on

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