pythagorean theorem:
[tex]a ^{2} + {b}^{2} = {c}^{2} [/tex]
there are 2 sides and a hypotenuse, a and b represent the sides and c is the hypotenuse, the hypotenuse is always the longest side to put it simply, now all you have to do is substitute, side a is missing, so we are solving for that missing side, side b is 9, and side c, the hypotenuse, is 12
[tex] {a}^{2} + {9}^{2} = {12}^{2} [/tex]
now simplify,
[tex] {a}^{2} + (9 \times 9) = (12 \times 12)[/tex]
[tex] {a}^{2} + 81 = 144[/tex]
now since we are solving for the a side, we must isolate a
[tex] {a}^{2} + 81 = 144 \\ - 81 \: \: \: \: \: \: \: \: - 81[/tex]
the 81's cancel out so it's just 144-81 which is 63
[tex] {a}^{2} = 63[/tex]
a still must be isolated, so we have to take the square root of both sides, that in turn cancels out the square on the a
[tex] \sqrt{a}^{2} = \sqrt{63} [/tex]
[tex]a = \sqrt{63} [/tex]
the equation can no longer be simplified so the answer is
[tex] a = \sqrt{63} [/tex]
