The legs of a right triangle have lengths $x + 2$ $x+2$ and $x + 6$ $x +6$ . The hypotenuse has length $2 x$ $2x$ . What is the perimeter of the triangle?

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The legs of a right triangle have lengths $x + 2$ $x+2$ and $x + 6$ $x +6$ . The hypotenuse has length $2 x$ $2x$ . What is the perimeter of the triangle?

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Answer 1 · 2018-01-08T00:05:19

The perimeter is $48$ $48$ . Note this is a constant, not depending on $x$ $x$ . We can solve for $x$ $x$ , see below.

Explanation:

Start with the Pythagorean Theorem:

Legs= $x + 2, x + 6$ $x+2, x+6$

Hypoteneuse= $2 x$ $2x$

Then

${(x + 2)}^{2} + {(x + 6)}^{2} = {(2 x)}^{2}$ $(x+2)^2+(x+6)^2=(2x)^2$

$(x^{2} + 4 x + 4) + (x^{2} + 12 x + 36) = 4 x^{2}$ $(x^2+4x+4)+(x^2+12x+36)=4x^2$

$2 x^{2} - 16 x - 40 = 0$ $2x^2-16x-40=0$

$2 (x + 2) (x - 10) = 0$ $2(x+2)(x-10)=0$

The sides of the triangle must be positive so $x = 10$ $color(blue)(x=10)$ .

Then the legs are $12$ $12$ and $16$ $16$ , the hypoteneuse is $20$ $20$ (check that $12^{2} + 16^{2} = 20^{2}$ $12^2+16^2=20^2$ ), and their sum is the perimeter $= 48$ $=48$ .

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The legs of a right triangle have lengths $x + 2$ $x+2$ and $x + 6$ $x +6$ . The hypotenuse has length $2 x$ $2x$ . What is the perimeter of the triangle?

1 Answer

Explanation:

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The legs of a right triangle have lengths $x + 2$ $x+2$ and $x + 6$ $x +6$ . The hypotenuse has length $2 x$ $2x$ . What is the perimeter of the triangle?