Aim: What’s so special about a 30 0 -60 0 -90 0 triangle?

Aim: Properties of Special Rt. Triangles Course: Alg. 2 & Trig.

Aim: What’s so special about a 300-600-900 triangle?

Do Now:

Triangle ABC is equilateral with each side equal to 2x. CD is an altitude of ABC.

What is mA? mB? mACB?

What is mACD? mBCD?

A B

C

2x 2x

2xD

600600600

600 600

300 300

300 300


300-600-900 triangle

Triangle ABC is equilateral with each side equal to 2x. CD is an altitude of ABC

What is length of CD in terms of x?

A B

C

2x 2x

2x

?Pythagorean Theorem - a2 + b2 = c2

x2 + (CD)2 = (2x)2

(CD)2 = 3x2

x x

x2 + (CD)2 = 4x2

D

600

300

CD 3x



Triangle ABC is equilateral with each side equal to 6 (instead of 2x). CD is an altitude of ABC

What is length of CD?

A B

C

6 6?

Pythagorean Theorem - a2 + b2 = c2

32 + (CD)2 = (6)2

(CD)2 = 27

3 3

9 + (CD)2 = 36D

600

300

CD 3 3

3 3

CD 27



A

C

2x

xD

3x

A

6

3

33

C

D

Review the results of the first two problems.

Can you make any general conclusions?

Problem 1

Problem 2

600

300

600

300



o o oIn a 30 -60 -90 triangle:

the length of the hypotenuse is twice the

length of the shorter leg.

the length of the longer leg is 3 times

the length of the shorter leg

The side opposite the 30 angle

is 1/2 the

length of the hypotenuse

o o oIn a 30 -60 -90 triangle:

the length of the hypotenuse is twice the

length of the shorter leg.

the length of the longer leg is 3 times

the length of the shorter leg

The side opposite the 30 angle

is 1/2 the

length of the hypotenuse

600

300

s

2s3s


45o - 45o - 90o triangle

Do Now:Triangle ABC is an isosceles right triangle with BC =

A. What is mB?

mC?

AB?

AC?

2 2

A B

C

2 2450

450


x

x

x2 + x2 = ( )22 2

2x2 = ( )22 22x2 = 8

2

2

= 2x


45o - 45o - 90o triangle

Do Now:Triangle ABC is an isosceles right triangle with BC =

A. What AB?

AC?

A B

C


x

x

x2 + x2 = ( )2

2x2 = 72

x2 = 36 x = 6

26

2

6

26

6

6



Review the results of the first two problems.

Can you make any general conclusions?

Problem 1

Problem 2

A B

C

2 2

2

2

A B

C

6

66 2


450- 450 - 900 triangle

s

In a 450-450-900 triangle, the length of the hypotenuse is times the length of a leg.

In a 450-450-900 triangle, the length of the hypotenuse is times the length of a leg.

450

450

s

Ratio of Hypotenuse : Leg of I.R.T is always

2s

2 : 1

2


Isosceles Right Triangle

450 – 450 - 900 triangle

B

F

E

I

H

D

G

CA

BFAB

= 1.4142 . . = 2

CGAC

= 1.4142 . . = 2

DHAD

= 1.4142 . . = 2

EIAE

= 1.4142 . . = 2

450

450

Ratio of Hypotenuse : Leg of I.R.T is always 2 : 1

cos45 2

sin45 2


Summary of Special Angles in Trig

0 30 45 60 90

1 2 3sin 0 1

2 2 2

3 2 1cos 1 0

2 2 2

3tan 0 1 3

3UND


Model Problem

Do Now:Triangle ABC is a 30-60-90 triangle with BC = 7

A. What is length of

AB?

AC?A B

C

7

600

300

Hypotenuse is 2 times the shorter leg

CB = 2(AB)

7 = 2(AB)

Longer leg is times the shorter leg3 3

AC = (AB) 3

AC 6.06

AC = (3.5)

3

3.5 = AB

3.5

3.5

3 3.5


Model Problem

Do Now:Triangle ABC is an isosceles right triangle with BC = 8

A. What AB?

AC?

A B

C

x

x8


x2 + x2 = (8)2

2x2 = 64

x2 = 32 x = 32

24 x =

Inste

ad of

4 2

4 2

Ratio of Hypotenuse : Leg of I.R.T is always 2 : 1

2 8

8

2

x

x

2

2

8 2

2x 4 2


Regents Prep

What is the exact sum of o 0sin60 cos90 ?

3

2+ 0

3

2


The rhombus below is a glass panel for a door. How many square inches of colored glass will you need for the panel?

Model Problem

6 in.

600600

6 in. 6 in.

Draw an altitude of the rhombus. Label x and h as shown x

h

6 = 2xHypotenuse is 2 times the shorter leg

3 = x

h = 3 3

Longer leg is times the shorter leg3 3

A = bh = 6( ) = 31.2 in23 3

A = bh


Model Problem

A baseball diamond is a square. The distance from base to base is 90 ft. To the nearest foot, how far does the second baseman throw a ball to home plate?

90’

90’

Isosceles Right Triangle

90 = 127.27922’2

hypotenuse is times the length of a leg.2

Documents

Aim: What’s so special about a 30 0 -60 0 -90 0 triangle?