SECTION 8.4 Area of a Triangle 579 Finding the Area of an SSS Triangle Find the area of a triangle whose sides are 4, 5, and 7. Solution EXAMPLE 2 Let = = a b 4, 5, and = c 7. Then ( ) ( ) = + + = + + = s a b c 1 2 1 2 4 5 7 8 Heron’s Formula gives the area K as ( )( )( ) ( )( )( ) = − − − = − − − = ⋅ ⋅ ⋅ = = K s s a s b s c 8 8 4 8 5 8 7 8 4 3 1 96 4 6 square units Now Work PROBLEMS 15 AND 23 Proof of Heron’s Formula The proof given here uses the Law of Cosines. From the Law of Cosines, = + − c a b ab C 2 cos 2 2 2 and the Half-angle Formula, = + C C cos 2 1 cos 2 2 it follows that ( ) ( )( ) ( ) ( ) = + = + + − = + + − = + − = + − + + = − ⋅ = − C C a b c ab a ab b c ab a b c ab a b c a b c ab s c s ab s s c ab cos 2 1 cos 2 1 2 2 2 4 4 4 2 2 4 2 2 2 2 2 2 2 2 2 (6) ↑ ↑ Difference of two squares ( ) + − = + + − = − = − a b c a b c c s c s c 2 2 2 2 Similarly, using = − C C sin 2 1 cos 2 , 2 it follows that ( )( ) = − − C s a s b ab sin 2 2 (7) Now use formula (2) for the area. ■ ( )( ) ( ) ( )( )( ) = = ⋅ = − − − = − − − K ab C ab C C ab s a s b ab s s c ab s s a s b s c 1 2 sin 1 2 2 sin 2 cos 2 ( ) = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = C C C C sin sin 2 2 2sin 2 cos 2 Use equations (6) and (7). Historical Feature Heron’s Formula (also known as Hero’s Formula) was first expressed by Heron of Alexandria (first century AD), who had, besides his mathematical talents, engineering skills. In various temples, his mechanical devices produced effects that seemed supernatural and supposedly moved visitors to generosity. Heron’s book Metrica, on making such devices, has survived and was discovered in 1896 in the city of Constantinople. Heron’s Formula for the area of a triangle caused some mild discomfort in Greek mathematics, because a product with two factors was an area and one with three factors was a volume, but four factors seemed contradictory in Heron’s time.
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