Answers:
x = sqrt(10)
y = sqrt(5)
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Step-by-step explanation:
Naturally I start with x as that letter precedes y in the alphabet; however, it's easier to start with y because it is a leg of this triangle. We will then use the value of y to find x later.
For any 45-45-90 triangle, the two legs are the same length. So that's why we're able to quickly see that y = sqrt(5)
To get the hypotenuse, we multiply the leg length by sqrt(2). This trick only works for 45-45-90 triangles.
hypotenuse = leg*sqrt(2)
x = sqrt(5)*sqrt(2)
x = sqrt(5*2)
x = sqrt(10)
The rule I used is sqrt(a)*sqrt(b) = sqrt(a*b)
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An alternate path is to use the pythagorean theorem to find x
a^2+b^2 = c^2
(sqrt(5))^2 + (sqrt(5))^2 = x^2
5 + 5 = x^2
10 = x^2
x^2 = 10
x = sqrt(10)