SOLUTION: If the area of a right isosceles triangle is 4, how long are its sides?

Algebra ->  Surface-area -> SOLUTION: If the area of a right isosceles triangle is 4, how long are its sides?      Log On


   

Question 697336: If the area of a right isosceles triangle is 4, how long are its sides?
Answer by Positive_EV(69) About Me  (Show Source):
You can put this solution on YOUR website!
The area of a triangle is equal to (1/2)*base*height. For a right triangle, the base and the height are the legs.
Since this is an isosceles triangle, both legs are the same length. Let L = the length of a leg:
%281%2F2%29%2AL%2AL+=+4 - divide both sides by 1/2:
L%5E2+=+8 - take the square root of both sides:
L+=+sqrt%288%29+=+2%2Asqrt%282%29.
The legs are both 2%2Asqrt%282%29. For the hypotenuse, you can use the Pythagorean theorem:
a%5E2+%2B+b%5E2+=+c%5E2, where a+=+b+=+2%2Asqrt%282%29:
%282%2Asqrt%282%29%29%5E2%2B%282%2Asqrt%282%29%29%5E2+=+c%5E2
8+%2B+8+=+c%5E2
16+=+c%5E2, take the square root of both sides:
4+=+c
The legs of the right triangle are 2%2Asqrt%282%29, and the hypotenuse is 4.
Follow-up edit: In this particular case, there are other ways to find the third side given the legs. Since this is a right isosceles (45-45-90) triangle, the ratios of the sides are going to be 1:1:sqrt%282%29.
So, another way to find the hypotenuse is to multiply the length of a leg by sqrt%282%29, which will also give 4 for the hypotenuse.
This specific method only works for right isosceles triangles, though. There's one other special case like this -- if you know the angles are 30, 60, and 90 degrees, the ratios of the sides are 1:sqrt%283%29:2. The Pythagorean theorem works on any right triangle -- if you know any two sides you can always find the third side of a right triangle with it.