ABC is a right angled triangle with ∠B=90∘. M is the midpoint of AC and BM=sqrt(117). Sum of the lengths of the other two sides AB and BC is 30. Find the area of the triangle ABC.
A
M
B C
The midpoint of the hypotenuse will be the circumcenter of a right triangle
Therefore AM = MC = sqrt (117)
So....the hypotenuse = 2sqrt (117) = sqrt (468)
Let AB = x so BC = 30 - x
And we have this equation
x^2 + (30 - x)^2 = 468 simplify
x^2 + x^2 - 60x + 900 = 468
2x^2 - 60x + 432 = 0
x^2 - 30x + 216 = 0
(x - 12) ( x -18) = 0
So depending upon your preference either AB = 12 and BC =18 or vice-versa
The area = (1/2) (product of the legs) = (1/2) 216 = 108