The legs of a right triangle have length x + 4 and x + 7. If the hypotenuse is 3x, what is the integral value of the perimeter?

The length of the legs of the right triangle is x + 4 and
x + 7. The hypotenuse is 3x. We also know that all the lengths are in integers, else the
perimeter cannot be an integer.

Use the Pythagorean
Theorem:

(x + 4)^2 + (x + 7)^2 =
(3x)^2

=> x^2 + 16 + 8x + x^2 + 49 + 14x =
9x^2

=> 2x^2 + 22x + 65 =
9x^2

=> 7x^2 - 22x - 65 =
0

=> 7x^2 - 35x + 13x - 65 =
0

=> 7x(x - 5) + 13(x - 5) =
0

=> (7x + 13)(x - 5) =
0

=> x = 5

the other
root is a non-integer and can be ignored

With x = 5, the
sides of the triangle are 9 , 12 , 15.

The perimeter is 9 +
12 + 15 = 36

The required perimeter of the
triangle is 36.