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**Please work on the pink warm up**

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**5.5 Special Triangles & Areas of Regular Polygons**

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

In a 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. In a triangle, the length of the hypotenuse is √2 times the length of the leg. x x √3 2x x √2 x FIND THE SHORT LEG FIRST!!!

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**LET’S TRY THIS… Short side first!! X = 6 Y = _____ Z = _____**

6 √3 12 Z 60 X 2 √3 X = _______ Y = Z = ________ 4 √3 30 Short side first!! Y P Q R 4 √2 8 P = ______ Q = 4√2 R= _____ 5 5√2 P = Q = ________ R = ________

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**Find the area game♥ A = ½ bh What is the length of the missing side? 6**

A = 18 sq. units 6 10 A = ½ bh Find the short side first!! A = ½ (5) (5√3) Find an exact answer A = 25√3 sq. units or √3 2 5√3 5

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**Areas of Regular Polygons**

What part of A=1/2bh is the perpendicular bisector? Can you find the area of a triangle? 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 apothem p is the perimeter of the polygon

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**Let’s see how this works…**

10 A = 1/2ap A = ½(6.88)(50) A = 172 sq.units 6.88 PAINLESS!! Let’s kick it up a notch…

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**Find the area of this one!**

Hmmmmm….. A circle has 360°… 12 Hmmmmm….. How many degrees would the top angle of each Δ have? 60° Hmmmmm….. Since the Δs are isosceles, what are the measures of the base angles? 30° 60° Hmmmmm….. If the apothem is an angle bisector, then what is the measure of the small top angle? 30° 60° The short side = 6 The apothem = 6√3 A = 1/2ap A = ½(6√3)(72) = 216√3 (exact) A = (approx.)

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WHEW! Try this… Find the perimeter and area of a Δ with a hypotenuse of 18units. (sketch it) 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 √3 units2 2 18 60° 9 30° 9√3 (hint: the smallest angle is across from the smallest side… the largest angle is across from the largest side.)

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Whiteboard time!!

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Assignment pg 336, 10-21

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