Question 692942: If the leg of an isosceles right triangle measures 12 in, then what is the length of the hypotenuse?
Found 2 solutions by jim_thompson5910, RedemptiveMath:
Answer by jim_thompson5910(35256)   (Show Source):
Answer by RedemptiveMath(80) (Show Source):
You can put this solution on YOUR website! An isosceles right triangle is a triangle with two sides that are equal. These equal sides are known as the legs. The other side, the longest side of any right triangle, is known as the hypotenuse. A right triangle can be operated on with a special theorem known as Pythagoras' theorem. It states that the sum of the square of the legs equals the square of the hypotenuse. This is true for any right triangle, and can usually only happen under right triangle conditions. So, let us look at the measures we are given.
We have a leg of an isosceles right triangle that equals 12 in. Since we know it is isosceles, the other leg must equal 12 in. as well. To find the the third side, the hypotenuse, we need to use Pythagoras' theorem to arrive there algebraically. Using "a" and "b" for the sides and "c" for the hypotenuse:
a^2 + b^2 = c^2
12^2 + 12^2 = c^2
144 + 144 = c^2
288 = c^2
c = [sqrt](288).
Depending on what course you are taking, this may or may not be an acceptable answer. In early math courses, this answer would be left as the square root of 288. Intermediate algebra courses would have this answer simplified using radical operations; however, I do not know how far you may be in algebra. So, if your teacher is fine with above answer, then you do not have to continue. If not, then we must simplify:
[sqrt](288) = [sqrt](144 * 2) = [sqrt](144) * [sqrt](2) = 12 * [sqrt](2).
You could also use trigonometric functions to solve for the hypotenuse. Taking the sine, cosine, cosecent or secant of an acute angle:
sin 45 deg = 12/x
{[sqrt](2)}/2 = 12/x
[sqrt](2) = 24/x
x * [sqrt](2) = 24
x = 24/{[sqrt](2)}
x = 24/{[sqrt](2)} * {[sqrt](2)}/{[sqrt](2)} = {24 * [sqrt](2)}/2 = 12 * [sqrt](2).
cos 45 deg = 12/x
cos 45 deg = sin 45 deg = {[sqrt](2)}/2
x = 12 * [sqrt](2).
csc 45 deg = x/12
[sqrt](2) = x/12
x = 12 * [sqrt](2).
sec 45 deg = 12/x
sec 45 deg = csc 45 deg = [sqrt](2)
x = 12 * [sqrt](2).
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