Question

The sides of a triangle are x , x +1 , 2 x -1 and its area is x root of 10 .Find the value of x.

Answer 1

Ok, I'm going to start off saying there is probably an easier way of doing this that's right in front of my face, but I can't see it so I'm going to use Heron's formula, which is A=√[s(s-a)(s-b)(s-c)] where A is the area, s is the semiperimeter (half of the perimeter), and a, b, and c are the side lengths.

Substitute the known values into the formula:

x√10=√{[(x+x+1+2x-1)/2][({x+x+1+2x-1}/2)-x][({x+x+1+2x-1}/2)-(x+1)][({x+x+1+2x-1}/2)-(2x-1)]}

Simplify:

x√10=√{[4x/2][(4x/2)-x][(4x/2)-(x+1)][(4x/2)-(2x-1)]}

x√10=√[2x(2x-x)(2x-x-1)(2x-2x+1)]

x√10=√[2x(x)(x-1)(1)]

x√10=√[2x²(x-1)]

x√10=√(2x³-2x²)

10x²=2x³-2x²

2x³-12x²=0

2x²(x-6)=0

2x²=0 or x-6=0

x=0 or x=6

Therefore, x=6 (you can't have a length of 0).