Special Right Triangles 45º-45º-90º

 

Certain triangles possess "special" properties that allow us to use "short cut formulas" in arriving at information about their measures.  These formulas let us arrive at the answer very quickly. One such triangle is the 45º-45º-90º triangle. There are two "special" formulas that apply ONLY to the 45º-45º-90º triangle.

 

45º-45º-90º (Isosceles Right Triangle)
"Special" Formulas

You must remember that these formulas can be used ONLY in a 45º-45º-90º triangle.

 

What if I forget the formulas? What should I do?

The nice thing about mathematics is that there is always another way to do the problem.  If you forget these formulas, you could always use the Pythagorean Theorem or a Trigonometry formula.

Let's look at 3 solutions to this problem where you are asked to find x:

Special Formula solution	Pythagorean Theorem solution	Trigonometric solution
We are looking for the hypotenuse so we will use the formula that will give the answer for the hypotenuse: Substituting the leg = 7, we arrive at the answer: A nice feature of these special formulas is that the answer is already in reduced form.	Since a 45º-45º-90º, also called an isosceles right triangle, has two legs equal, we know that the other leg also has a length of 7 units. c2 = a2 + b2 x2 = 72 +72 x2 = 49 + 49 x2 = 98	Use either 45º angle as the reference angle (where your stick figure will stand).  One possible solution is shown below: rounded rounded