25.5 Special Triangles & Areas of Regular Polygons
3There 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.xx √32xx √2xFIND THE SHORT LEG FIRST!!!
4LET’S TRY THIS… Short side first!! X = 6 Y = _____ Z = _____ 6 √312Z60X2 √3X = _______ Y = Z = ________4 √330Short side first!!YPQR4 √28P = ______ Q = 4√2 R= _____55√2P = Q = ________ R = ________
5Find the area game♥ A = ½ bh What is the length of the missing side? 6 A = 18 sq. units610A = ½ bh Find the short side first!!A = ½ (5) (5√3)Find an exact answerA = 25√3 sq. units or √325√35
6Areas 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 = ½apa is the length of the apothemp is the perimeter of the polygon
7Let’s see how this works… 10A = 1/2apA = ½(6.88)(50)A = 172 sq.units6.88PAINLESS!!Let’s kick it up a notch…
8Find the area of this one! Hmmmmm….. A circle has 360°…12Hmmmmm….. 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 = 6The apothem = 6√3A = 1/2apA = ½(6√3)(72) = 216√3 (exact)A = (approx.)
9WHEW! 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 unitsWhat is the area? (exact)81√3 sq units or √3 units221860°930°9√3(hint: the smallest angle is across from the smallest side… the largest angle is across from the largest side.)