Find the area of the triangle.
The side lengths of
△
A
B
C
whose perimeter is
7
6
, are
2
x
+
1
2
,
2
4
and
1
6
+
6
x
; where
x
is a variable. Find the area of the triangle.
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2 solutions
P = s u m o f s i d e l e n g t h s
7 6 = 2 x + 1 2 + 2 4 + 1 6 + 6 x
8 x = 2 4
x = 3
So the other sides are 2 x + 1 2 = 2 ( 3 ) + 1 2 = 1 8 and 1 6 + 6 x = 1 6 + 6 ( 3 ) = 3 4 .
Using the Heron’s Formula, the area is
s = 2 1 8 + 2 4 + 3 4 = 3 8
A = s ( s − a ) ( s − b ) ( s − c ) = 3 8 ( 3 8 − 1 8 ) ( 3 8 − 2 4 ) ( 3 8 − 3 4 ) = 4 2 5 6 0 = 8 6 6 5
In case you wish to avoid using Heron's formula, you may assume C:(0,0), B:(34,0) and A:(p,q). Then, p²+q²=18²,(p-34)²+q²=24² which yield: p = 226/17, q = 8√665/17 and A = (1/2)(17)*q = 8√665