Special right triangles

There are two types of special right triangles and they have the following properties:

The first one is the 45°-45°-90° triangle

45-45-90 degrees special right 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

Hypothenuse2 = Leg2 + Leg2

Hypothenuse2 = 2×Leg2

Let's take the square root of both sides

√(Hypothenuse2) = √(2×Leg2)

Hypothenuse = √(2)×√(Leg2)

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

30-60-90 degrees special right triangle
Since the Short Leg is 1/2 the Hypothenuse, the Hypothenuse is 2 × Short Leg

Using the pythagorean theorem, we get

Hypothenuse2 = (Short Leg)2 + (Long Leg)2

(2 × Short Leg)2 = (Short Leg)2 + (Long Leg)2

(2 × Short Leg)×(2 × Short Leg) = (Short Leg)2 + (Long Leg)2

(4 × (Short Leg)2) = (Short Leg)2 + (Long Leg)2

4 × (Short Leg)2 − (Short Leg)2 = (Short Leg)2 − (Short Leg)2 + (Long Leg)2

4 × (Short Leg)2 − (Short Leg)2 = 0 + (Long Leg)2

4 × (Short Leg)2 − (Short Leg)2 = (Long Leg)2

3 × (Short Leg)2 = (Long Leg)2

Let's take the square root of both sides

√(3 × Short Leg2) = √((Long Leg)2)

(√3) × √((Short Leg)2) = Long Leg

(√3) × (Short Leg) = Long Leg

So, the formula for the long leg is: Long Leg = (√3) × (Short Leg)

Since the short leg = 1/2(Hypothenuse), Long Leg is also equal to:

(√3) × 1/2(Hypothenuse)

Example #2:

The Hypothenuse of a 30°-60°-90° triangle is equal to 20 inches. Find the Short Leg and the Long Leg.

Short Leg = 1/2 × (Hypothenuse) = 1/2 × 20 = 10 inches

Long leg = (√3) × Short Leg = √3) × 10 = 17.32 inches

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