The hypotenuse of a right triangle is 10 inches. The lengths of the two legs are given by 2 consecutive even integers. How do you find the lengths of the two legs?

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The hypotenuse of a right triangle is 10 inches. The lengths of the two legs are given by 2 consecutive even integers. How do you find the lengths of the two legs?

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Answer 1 · 2016-01-04T02:48:15

$6, 8$ $6,8$

Explanation:

The first thing to tackle here is how to express "two consecutive even integers" algebraically.

$2 x$ $2x$ will give an even integer if $x$ $x$ is also an integer. The next even integer, following $2 x$ $2x$ , would be $2 x + 2$ $2x+2$ . We can use these as the lengths of our legs, but must remember that this will only hold true if $x$ $x$ is a (positive) integer.

Apply the Pythagorean theorem:

${(2 x)}^{2} + {(2 x + 2)}^{2} = 10^{2}$ $(2x)^2+(2x+2)^2=10^2$

$4 x^{2} + 4 x^{2} + 8 x + 4 = 100$ $4x^2+4x^2+8x+4=100$

$8 x^{2} + 8 x - 96 = 0$ $8x^2+8x-96=0$

$x^{2} + x - 12 = 0$ $x^2+x-12=0$

$(x + 4) (x - 3) = 0$ $(x+4)(x-3)=0$

$x = - 4, 3$ $x=-4,3$

Thus, $x = 3$ $x=3$ since the side lengths of the triangle can't be negative.

The legs are

$2 x \Rightarrow 6$ $2xrArr6$
$2 x + 2 \Rightarrow 8$ $2x+2rArr8$
$hypotenuse \Rightarrow 10$ $"hypotenuse"rArr10$

A more intuitive way to do this problem is to recognize that a $6, 8, 10$ $6,8,10$ triangle is just twice the size of the fundamental $3, 4, 5$ $3,4,5$ right triangle.

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The hypotenuse of a right triangle is 10 inches. The lengths of the two legs are given by 2 consecutive even integers. How do you find the lengths of the two legs?

1 Answer

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