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` .

`x^2+x^2+14x+49=169`

Then, combine like terms.

`2x^2+14x+49=169`

Express the equation in quadratic form `ax^2+bx+c =0` . So move 169 to the left side.

`2x^2+14x+49-169=0`

`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.