Answer: A) m = 9, n = 9
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Explanation:
This is a 45-45-90 right triangle. Any triangle of this form is isosceles, where the two legs are the same length. In this case, m and n are those two legs. So m = n.
We can then apply the Pythagorean theorem to say
a^2 + b^2 = c^2
m^2 + n^2 = (9*sqrt(2))^2
m^2 + m^2 = 9^2*(sqrt(2))^2
2m^2 = 81*2
m^2 = (81*2)/2
m^2 = 81
m = sqrt(81)
m = 9
As a shortcut, the hypotenuse of a 45-45-90 triangle is equal to sqrt(2) times the length of either leg. If m is the length of the leg, then m*sqrt(2) is the length of the hypotenuse. So in a sense, we take this thought process backwards to go from a hypotenuse 9*sqrt(2) to a leg length of 9.
