A triangle has sides of 12 m, 16 m, and 22 m. Find the area of the triangle, giving your answer to three significant figures.
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Solution 1: Heron's Formula
Let the sides of the triangle be a = 1 2 , b = 1 6 and c = 2 2 . Then the semi-perimeter is
s = 2 a + b + c = 2 1 2 + 1 6 + 2 2 = 2 5
So the area is
A = s ( s − a ) ( s − b ) ( s − c ) = 2 5 ( 2 5 − 1 2 ) ( 2 5 − 1 6 ) ( 2 5 − 2 2 ) ≈ 9 3 . 7
Solution 2: Cosine Rule
Let the sides of the triangle be a = 1 2 , b = 1 6 and c = 2 2 and let the opposite angle of side a be A , of side b be B and of side c be C . By cosine rule, we have
c 2 = a 2 + b 2 − 2 a b cos C
2 2 2 = 1 2 2 + 1 6 2 − 2 ( 1 2 ) ( 1 6 ) cos C
C ≈ 1 0 2 . 6 4 ∘
So the area is
A = 2 1 a b sin C = 2 1 ( 1 2 ) ( 1 6 ) ( sin 1 0 2 . 6 4 ) ≈ 9 3 . 7