There are two types of special right triangles and they have the following properties:
The first one is the 45^{°}-45^{°}-90^{°} triangle
In this triangle, the important thing to remember is that the legs have equal length.
Let's call the longest side Hypothenuse. Using the pythagorean theorem, we get
Hypothenuse^{2} = Leg^{2} + Leg^{2}
Hypothenuse^{2} = 2×Leg^{2}
Let's take the square root of both sides
√(Hypothenuse^{2}) = √(2×Leg^{2})
Hypothenuse = √(2)×√(Leg^{2})
So, the formula is:
Hypothenuse = √(2)×(Leg)
Now, what's the point of having a formula like that?
Well, it is a shortcut to solve problems. If the legs of a right triangle are equal, you can quickly find the length of the legs or the hypothenuse
given the length of the hypothenuse or the length of a leg respectively.
Example #1:
The legs of an isosceles right triangle measure 10 inches. Find the length of the hypothenuse.
Since the triangle is isosceles, the legs are equal and we can use the formula
Hypothenuse = √(2)×(Leg)
Hypothenuse = √(2)×(10)= 14.1421 inches
The second type of special right triangles is the 30^{°}-60^{°}-90^{°} triangle
Since the Short Leg is 1/2 the Hypothenuse, the Hypothenuse is 2 × Short Leg