Unit 6 - Right Triangles

Radical sign

aa

Radicand

Index

b

Simplifying Radicals

04/20/2023Algebra Review

4

9

16

25

36

49

64

Perfect Squares

Simplifying Radicals

81

100

121

144

169

196

225

9 2

=

10 2 =

11 2 =

12 2 =

13 2 =

14 2 =

15 2 =

2 2

=

3 2 =

4 2 =

5 2 =

6 2 =

7 2 =

8 2 =

04/20/2023Algebra Review

4 = 2

Their square roots

81 = 9

100 = 10

121 = 11

169 = 13

144 = 12

196 = 14

225 = 15

16 = 4

25 = 5

36 = 6

64 = 8

49 = 7

9 = 3

Simplifying Radicals

04/20/2023Algebra Review

12

Example 11. Find the largest perfect square that will into the radicand evenly.

2. Write the radicand in factored form using the perfect square as a factor.

3. Simplify the perfect square. (Remove it from the radical.)

Simplifying Radicals

04/20/2023Algebra Review

48

Example 2

2. Write the radicand in factored form using the perfect square as a factor.

3. Simplify the perfect square. (Remove it from the radical.)

Simplifying Radicals

1. Find the largest perfect square that will into the radicand evenly.

04/20/2023Algebra Review

54 311

73 40 512

80

Example 3

Example 4 363

Example 5 63

Example 6

Example 7160

0720

Simplifying Radicals

04/20/2023Algebra Review

5514

106 159

400

Example 8

Example 9 10780

Example 10

360

Example 11

Example 12

3185

1215

20

657

Simplifying Radicals

04/20/2023Algebra Review

1. No perfect square radicand2. No perfect square factor of the radicand3. No fraction as a radicand4. No radical in the denominator

5. No unreduced fractions

Rules for Simplifying Radicals

04/20/2023Algebra Review

Geometric Mean

x is said to be the geometric mean between a and b if and only if :

a x x b

=

Right Triangles 04/20/2023

Example 1

Find the geometric mean between 4 and 25.

x x

=25

4

X 2 = 100

X = 100 = 10

10 is the geometric mean between 4 & 25.

Right Triangles 04/20/2023

Example 2Find the geometric mean between 3 and 6.

x

x=

63

X 2 = 18

X = 18 = 9 · 2 = 3 2

Right Triangles 04/20/2023

Right Triangles 04/20/2023

1. Cut an index card along one of its diagonals.

2. On one of the right triangles, draw an altitude from the right angle to the hypotenuse.

3. Cut along the altitude to form two smaller right triangles.You should now have three right triangles.

Compare the triangles. What special property do they share? Explain.

part 2part 2part 1part 1

If the altitude is drawn to the hypotenuse

in a right triangle, then the length of the

altitude is the geometric mean between

the lengths of the parts of the hypotenuse.

altaltalt

altpart 1=

part 2

Right Triangles 04/20/2023

part 2part 2part 1part 1

If the altitude is drawn to the hypotenuse in a right triangle, then the length of a leg is the geometric mean between the length of the part of the hypotenuse adjacent to the leg and the length of the hypotenuse.

Leg 1part 1

=hyp

Leg 2

hypotenuse

Leg 1

Leg 1

part 2=

hypLeg 2

Leg 2

altalt

Right Triangles 04/20/2023

4 3

x zzy

Example 3

Find the value of

x, y, and z.

Right Triangles 04/20/2023

c 2 = 100

The Pythagorean Theorem

The square of the length of the hypotenuse in a right triangle is equal to the sum of the squares of the lengths of the legs.

c 2 = a 2 + b 2

c 2 = 6 2 + 8 2

c = 100 = 10

c

b

a

8

6c 2 = 36 + 64

EX. 4

Right Triangles 04/20/2023

- 36 - 36

c 2 = a 2 + b 2

12 2 = a 2 + 6 2

108 = a 2

c

b

a12

6144 = a 2 + 36

EX. 5

a = 108 = 36·3 = 6 3

Right Triangles 04/20/2023