Solving for the Legs of a 45°-45°-90° Triangle Using the Hypotenuse

Introduction to 45°-45°-90° Triangles

Understanding the properties of a 45°-45°-90° triangle is crucial in geometry and trigonometry. This specific type of triangle features two equal legs and a hypotenuse that is √2 times the length of each leg. Such properties are widely used in various real-world applications, from construction to engineering.

Key Properties of a 45°-45°-90° Triangle

For a 45°-45°-90° triangle, the ratios of the side lengths are as follows:

The legs are equal. The hypotenuse is √2 times the length of each leg.

If the hypotenuse (H) is known, the length of each leg (L) can be calculated using the formula:

H L√2

Given Hypotenuse: Solving for the Leg

Let's work through a specific example. Suppose the hypotenuse (H) of a 45°-45°-90° triangle is 6 units. The goal is to determine the length of each leg (L).

Step-by-Step Solution

1. Start with the formula relating the hypotenuse and the leg:

H L√2

2. Substitute the given value of the hypotenuse (H) into the formula:

6 L√2

3. Solve for L by isolating the variable:

L 6 / √2

4. Simplify the expression by rationalizing the denominator:

L 6√2 / 2

5. Simplify further:

L 3√2

Verification Using Trigonometric Ratios

Alternatively, we can verify this solution using trigonometric ratios. The sine of 45° is given by:

sin(45°) L / H

Given that H is 6 units:

sin(45°) L / 6

Knowing that sin(45°) 1/√2:

1/√2 L / 6

Solving for L:

L 6 / √2

Rationalizing the denominator once again:

L 6√2 / 2 3√2

Concluding Notes and Additional Examples

Understanding the relationship between the hypotenuse and legs of a 45°-45°-90° triangle is foundational for more advanced topics in mathematics and problem-solving. Memorizing the key ratios (1 : 1 : √2) will greatly simplify solving related problems.

Let us consider another example: If the hypotenuse is 8 units, then each leg (L) would be:

L 8 / √2 8√2 / 2 4√2

By following similar steps and processes, you can solve for the legs of any 45°-45°-90° triangle given its hypotenuse.

Conclusion

In conclusion, the hypotenuse of a 45°-45°-90° triangle is always √2 times the length of each leg. This relationship is a fundamental principle in geometry and trigonometry, providing a quick and efficient way to solve for either the hypotenuse or the legs of a triangle, given one of the sides.