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
mason m
Jan 4, 2016

#6,8#

Explanation:

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

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

Apply the Pythagorean theorem:

#(2x)^2+(2x+2)^2=10^2#

#4x^2+4x^2+8x+4=100#

#8x^2+8x-96=0#

#x^2+x-12=0#

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

#x=-4,3#

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

The legs are

#2xrArr6#
#2x+2rArr8#
#"hypotenuse"rArr10#

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

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