# 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?

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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.**

Given the sides of the right angle triangle are x+ 4, x+ 7, and the hypotenuse is 3x.

Then :

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

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

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

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

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

==> x = -13/7 ( impossible the length can not be negative.)

==> x= 5

Then the sides are:

5+4 , 5+ 7, 3*5 ==> 9, 12, 15

Then the perimeter is :

P = 9 + 12 + 15 = 36

**Then the value of the perimeter is 36 units.**