Since we have a right triangle, apply the Pythagorean formula to solve for x. The formula is:
`a^2 + b^2 = c^2`
where c is the sloping side or the hypotenuse of the triangle, and a and b are the other two sides of the right triangle which corresponds to the height and the base.
So we have,
`x^2 + (x+7)^2 = 13^2`
`x^2 + (x+7)^2 = 169`
Expand `(x+7)^2` .
Then, combine like terms.
Express the equation in quadratic form `ax^2+bx+c =0` . So move 169 to the left side.
`2x^2+14x-120 = 0`
Since 2, 14 and -120 are all divisible by 2, divide both sides by 2.
`(2x^2+14x-120)/2 = 0/2`
`x^2+7x - 60 = 0`
Then, factor left side.
`(x-5)(x+12) = 0`
To solve for x, set each factor equal to zero.
`x-5=0` and `x+12=0`
`x=5 ` `x=-12`
Since x represent the height of the triangle, consider only the positive value of x.
Hence, the height of the right triangle is 5.