Please work on the pink warm up

5.5 Special Triangles &

Areas of Regular Polygons

There are 2 special right triangles 45-45-90 30-60-90

In a 45-45-90 triangle, the length of the hypotenuse is √2 times the length of the leg.

x √2

x

x

In a 30-60-90 triangle, the

length of the hypotenuse is 2 times the length of the shorter leg AND the length

of the longer leg is √3 times the

length of the shorter leg.

FIND THE SHORT LEG FIRST!!!FIND THE SHORT LEG FIRST!!!

x

x √32x

LET’S TRY THIS…X = 6 Y = _____ Z = _____

60

30

X

Y

Z

6 √3 12

X = _______ Y = 6 Z = ________4 √32 √3

Short side first!!

P = ______ Q = 4√2 R= _____

P = 5 Q = ________ R = ________

4 √2 8

5 5√2

P

Q

R

Find the area game♥

A = ½ bh What is the length of the missing side?

6A = ½ (6) (6)A = 18 sq. units

10

A = ½ bh Find the short side first!!

6

5

5√3A = ½ (5) (5√3)Find an exact answerA = 25√3 sq. units or 12.5√3 2

Areas of Regular Polygons

The perpendicular bisector of a triangle in a polygon is called an APOTHEM.

The formula for the area of a regular polygon is:

A = ½ap

a is the length of the apothemp is the perimeter of the polygon

Can you find the area of a triangle?

What part of A=1/2bh is the perpendicular bisector?

Let’s see how this works…

A = 1/2ap

A = ½(6.88)(50)

A = 172 sq.units

10

6.88

PAINLESS!!

Let’s kick it up a notch…

Find the area of this one!12

Hmmmmm….. A circle has 360°…

Hmmmmm….. How many degrees would the top angle of each Δ have?

Hmmmmm….. Since the Δs are isosceles, what are the measures of the base angles?

Hmmmmm….. If the apothem is an angle bisector, then what is the measure of the small top angle?

60°

60°60°

30°

30°

60°

The short side = 6The apothem = 6√3A = 1/2ap A = ½(6√3)(72) = 216√3 (exact)A = 374.12 (approx.)

WHEW!Try this…

Find the perimeter and area of a 30-60-90 Δ with a hypotenuse of 18units.

(sketch it)

18

(hint: the smallest angle is across from the smallest side… the largest angle is across from the largest side.)

30°

60°

What is the length of the short side?

What is the length of the long leg?

What is the perimeter? (exact)

27 + 9√3 units

What is the area? (exact)81√3 sq units or 40.5√3 units2

2

9

9√3

Whiteboard time!!

Assignment

pg 336, 10-21