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 hypotenuse. Using the Pythagorean theorem, we get:

Hypotenuse2 = Leg2 + Leg2

Hypotenuse2 = 2×Leg2

Let's take the square root of both sides

√(Hypotenuse2) = √(2×Leg2)

Hypotenuse = √(2)×√(Leg2)

So, the formula is:

Hypotenuse = √(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 a leg if the length of the hypotenuse is given or the length of the hypotenuse if the length of a leg is given.

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 hypotenuse, the hypotenuse is 2 × short leg.

Using the Pythagorean theorem, we get:

Hypotenuse2 = (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(hypotenuse), long Leg is also equal to:

(√3) × 1/2(hypotenuse)

A couple of problems about special right triangles

Example #1:

The legs of an isosceles right triangle measure 10 inches. Find the length of the hypotenuse.

Since the triangle is isosceles, the legs are equal and we can use the formula.

Hypotenuse = √(2)×(Leg)

Hypotenuse = √(2)×(10)= 14.1421 inches

Example #2:

The hypotenuse of a 30°-60°-90° triangle is equal to 20 inches. Find the short leg and the long leg.

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

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

Send me questions here about special right triangles.

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