45-45-90 Triangles: A Guide to Finding Missing Sides

In the world of geometry, understanding triangles is crucial. Among the various types of triangles, 45-45-90 triangles hold a special place due to their unique properties. These triangles are characterized by having two equal angles of 45 degrees each and a right angle of 90 degrees. The special relationship between their sides makes them particularly useful in solving problems involving right triangles.

Understanding the Special Ratio

The key to solving 45-45-90 triangle problems lies in understanding the special ratio of their sides. Here’s the breakdown:

• Hypotenuse: The side opposite the right angle is called the hypotenuse. It is always the longest side of the triangle.
• Legs: The two sides that form the right angle are called the legs. In a 45-45-90 triangle, the legs are equal in length.

The ratio of the sides in a 45-45-90 triangle is always 1:1:√2. This means:

• The hypotenuse is √2 times the length of each leg.
• Each leg is equal to the hypotenuse divided by √2.

Using the Pythagorean Theorem

You can also use the Pythagorean Theorem to find missing sides in a 45-45-90 triangle. The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b):

a² + b² = c²

Since the legs of a 45-45-90 triangle are equal, we can simplify this equation to:

2a² = c²

Examples

Example 1: Finding the Hypotenuse

Let’s say you have a 45-45-90 triangle where each leg is 5 units long. To find the hypotenuse, you can use the special ratio:

Hypotenuse = √2 * Leg = √2 * 5 = 5√2 units

You can also use the Pythagorean Theorem:

2a² = c²

2 * 5² = c²

50 = c²

c = √50 = 5√2 units

Example 2: Finding a Leg

Suppose you have a 45-45-90 triangle with a hypotenuse of 8 units. To find the length of each leg, you can use the special ratio:

Leg = Hypotenuse / √2 = 8 / √2 = 4√2 units

You can also use the Pythagorean Theorem:

2a² = c²

2a² = 8²

2a² = 64

a² = 32

a = √32 = 4√2 units

Practice Problems

Here are some practice problems to test your understanding:

1. A 45-45-90 triangle has a leg length of 7 units. What is the length of the hypotenuse?
2. A 45-45-90 triangle has a hypotenuse length of 10 units. What is the length of each leg?
3. A 45-45-90 triangle has a leg length of √2 units. What is the length of the hypotenuse?

Conclusion

Understanding 45-45-90 triangles and their special ratio is essential for solving problems involving right triangles. By applying the Pythagorean Theorem and the special ratio, you can efficiently find the missing sides of these triangles. Practice these concepts, and you’ll be well on your way to mastering geometry!