Geometry

8.1 Right Triangles

A radical is in simplest form when:

1. No perfect square factor other than 1 is under the radical sign.2. No fraction is under the radical sign.3. No radical is in a denominator.

Simplify:

5 2501. 81 2. 24 3. 300 4. 5. 6. 4 27

483

15 5 301. 9 2. 2 6 3. 10 3 4. 5. 6. 12 3

3 12

Answers:

#2 and #4 Together…You got the rest!!

Geometric MeanIf a, b and x are positive numbers and

ax

x=bmean

mean

Multiplying means/extremes we find that:

then x is called the geometric mean between a and b.

2x =abTaking the square root of each side: abx=

A positive number

Geometric Mean• Basically, to find the geometric mean of 2 numbers,

multiply them and take the square root. OR (add to notes: take the square root of each, then multiply try to find the geometric mean between 4 and 9 and the geometric mean between 36 and 50)

• Note that the geometric mean always falls between the 2 numbers.

abx=

Ex: Find the geometric mean between 5 and 11.

Use the formula

5 11 = 55

Geometric Mean

5 20 = 100 =10

abx=Ex: Find the geometric mean between the two numbers.

Avoid multiplying large numbers together. Break numbers into perfect square factors to simplify.

a. 5 and 20

b. 24 and 32

24 32 = 24 32

= 4 6 2 16 = 2 6 4 2

=8 12 =8 4 3 =16 3

Directions: Find the geometric mean between the two numbers.

8. 64 and 49 9. 1 and 3 10. 100 and 6 11. 20 and 24

Geometric Mean

64 49 =8 7 =56

1 3 = 3

5 20 = 100 =10

abx=Find the geometric mean between the two numbers.

7. 5 and 20

8. 64 and 49

9. 1 and 3

10.100 and 6

11.20 and 24

100 6 =10 6

20 24 =2 5 2 6 =4 30

∆ Review

• Altitude the perpendicular segment from a vertex to the line containing the opposite side

Hypotenuse the side opposite the right angle in a right triangle

Theorem

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.

~~

a

b

a

b

Corollaries

When the altitude is drawn to the hypotenuse of a right triangle:

the length of the altitude is the geometric mean between the segments of the hypotenuse.

Each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

Y

A ZX

Corollary 1

piece of hypotenuse altitude altitude other piece

of hypotenuse

Y

A ZX

=

XA YA=

YA AZ

Corollary 2

hypotenuse leg leg piece of

hyp. adj. to leg

Y

A ZX

=

XZ XYFor legXY: =

XY XA

Corollary 2

Y

A ZX

XZ YZFor legYZ: =

YZ AZ

hypotenuse leg leg piece of

hyp. adj. to leg

=

Directions: Exercises 24-31 refer to the diagram at right.

12. If CN = 8 and NB = 16, find AN.

13. If AN = 4 and CN = 12, find NB.

14. If AN = 4 and CN = 8, find AB.

15. If AB = 18 and CB = 12, find NB.

16. If AC = 6 and AN = 4, find NB.N

A

C

B

Homework

pg. 288 #16-30, 31-39 odd

ExercisesC

N BA

12. If CN = 8 and NB = 16, find AN. Let x = AN

8

16

2

8

8 16

8 16 64 16

4

long leg x

short leg

x x

x

x

ExercisesC

N BA

14. If AN = 8 and NB = 12, find CN. Let x = CN

x

12

2

8

12

96 96 6 16

4 6

long leg x

short leg x

x x

x

8

ExercisesC

N BA

17. If AB = 18 and CB = 12, find NB. Let x = NB

x

12

2

. 18 12

12

12 18 144 18

8

hypot

short leg x

x x

x

18

Answers Exercises 13 - 19

13. 3615. 2016. 2√1518. 2519. 5