I need to know how to figure the length of 2 sides of a triangle, as the following example

The length of the bottom is 12' and the angles are 45, 45 I need to know how long the other 2 sides must be to get an angle of 90 at the top.

Secondary Level
Other than student

Thank you,



Hi Lonnie.

If you construct a triangle where you know two angles, you automatically know the third, because it is impossible to construct a triangle whose angles don't total 180 degrees. So if you have two angles of 45 degrees each, the top MUST be 90 degrees or you don't have a triangle at all.

Additionally, any triangle where two angles are equal sizes must have the sides opposite those angles the same. We call these triangles "isosceles" triangles.

Any triangle (like yours) where one angle is 90 degrees is called a "right triangle". All right triangles have an interesting property discovered by the ancient Greeks that we now call the Pythagorean Theorem. This lets us calculate the length of one side if we know the length of two others. It says that if c is the length of the hypotenuse (the side opposite the 90 degree angle) and a and b are the lengths of the other two sides, then a2 + b2 = c2.

In your case we have a right isosceles triangle. This means that lengths a and b are the same. So we can substitute that into the pythagorean equation above to get a2 + a2 = c2. That simplifies to 2a2 = c2. Since you know the length opposite the right triangle is 12', then

2a2 = 122
a2 = 144/2
a = sqrt(72)
a = (roughly) 8.485 feet = (roughly) 101 3/4 inches.

Hope this helps,
Stephen La Rocque.