Understanding 45°-45°-90° Right Triangles with Given Hypotenuse

In a right triangle where one of the angles is 45°, the triangle is an isosceles right triangle. This unique type of triangle means that the two legs are of equal length. Let's delve into the specifics of such a triangle and explore how to find the lengths of the other two sides given the length of the hypotenuse.

The Mathematical Setup

The Pythagorean theorem is a foundational principle in geometry and trigonometry, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is represented as:

c^2 a^2 b^2

Given that the triangle is isosceles, we denote the two equal legs as a and b, so a b. Substituting this into the Pythagorean theorem, we can simplify our equation as:

c^2 a^2 a^2 2a^2

Applying the Given Example

Suppose the length of the hypotenuse c is 3√2 inches. We can now substitute this value into our simplified equation:

(3√2)^2 2a^2

Note: Recall that (a√b)^2 a^2 * b.

Calculating the left side:

9 * 2 18

Therefore, we have:

18 2a^2

Next, we divide both sides by 2:

9 a^2

Finally, taking the square root of both sides:

a 3

Hence, the lengths of the other two sides (the legs of the triangle) are each 3 inches.

Finding the Side Length in a Simplistic Calculation

A simpler way to approach this would be through the properties of a 45°-45°-90° triangle. Since it is isosceles and one of the angles is 45°, the other angle must also be 45°, making the triangle isosceles. If the hypotenuse is represented as H, then the legs will each be H/√2. For our case:

H 3√2

Side (3√2) / √2 3 inches

Conclusion

To summarize, in a right triangle with a 45° angle, the length of each leg is equal, and the hypotenuse is √2 times the length of one leg. By following the steps of the Pythagorean theorem or leveraging the geometric properties of a 45°-45°-90° triangle, we can quickly find the lengths of the sides given one side length, making it an essential tool for both theoretical and practical applications in mathematics and engineering.