The legs of a right triangle have lengths of x + 4 and x + 7. The hypotenuse length is 3x. How do you find the perimeter of the triangle?

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Answer 1 · 2016-01-03T19:50:57

#36#

The perimeter is equal to the sum of the sides, so the perimeter is:

#(x+4)+(x+7)+3x=5x+11#

However, we can use the Pythagorean theorem to determine the value of #x# since this is a right triangle.

#a^2+b^2+c^2#

where #a,b# are legs and #c# is the hypotenuse.

Plug in the known side values.

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

Distribute and solve.

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

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

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

Factor the quadratic (or use the quadratic formula).

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

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

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

#x=-13/7,5#

Only #x=5# is valid here, since the hypotenuse's length would be negative if #x=-13/7#.

Since #x=5#, and the perimeter is #5x+11#, the perimeter is:

#5(5)+11=36#