+1439 Triangle $ABC$ is isosceles with $AB = BC.$ If $AC = 20$ and $[ABC] = 100 \sqrt{3},$ then find the perimeter of triangle $ABC.$
+129883 B
A 20 C
The altitude is
100 sqrt (3) = (1/2) (20) (altitude)
100 sqrt (3) = 10 (altitude) divide both sides by 10
10sqrt (3) = altitude = sqrt (300)
Call the altitude BD
DC = 10
BC= ????
Since the triangle issoceles, the altitude bisects the base
So by the Pythagorean Theorem
VC = sqrt ( DC^2 + BD^3)
BC = sqrt ( 10^2 + (sqrt 300)^2 ] = sqrt [ 400] = 20
The perimeter = AB + BC + AC = 20 + 20 + 20 = 60
