Solving for the Length of a Leg in a 45°-45°-90° Triangle Using Pythagoras: A Comprehensive Guide

When faced with the task of finding the length of the legs in a 45°-45°-90° triangle given the hypotenuse, many methods can be used, such as the Pythagorean theorem. Letrsquo;s walk through a detailed process to solve the problem: if the hypotenuse of a 45°-45°-90° triangle is 13, find the length of one of the legs.

Using the Pythagorean Theorem

The Pythagorean theorem states that:

For any right triangle, (a^2 b^2 c^2)

In a 45°-45°-90° triangle, the two legs are of equal length. Let each leg be (a). Applying the Pythagorean theorem:

(a^2 a^2 h^2)

(2a^2 h^2)

(2a^2 13^2)

(2a^2 169)

(a^2 frac{169}{2})

(a^2 84.5)

(a sqrt{84.5})

(a approx 9.2) (rounded to one decimal place)

However, upon re-evaluating the initial input, we see that the original input specifies 13 as the hypotenuse which leads to:

(2a^2 169)

(a^2 frac{169}{2})

(a^2 84.5)

(a sqrt{84.5} approx 9.213)

Alternatively, using the simplified form:

(2a^2 169) leads to(a^2 frac{169}{2} 84.5) and(a sqrt{84.5} approx 9.213)

This confirms that each leg is approximately 9.213 units.

Isosceles Right Triangle Property

A 45°-45°-90° triangle, also known as an isosceles right triangle, has two equal legs and one hypotenuse. This property can be used to simplify the calculations:

Let each equal side be (x).

(x^2 x^2 10^2)

(2x^2 100)

(x^2 frac{100}{2} 50)

(x sqrt{50} 5sqrt{2}) units

Thus, each leg is (5sqrt{2}) units.

Alternative Methods

Other methods to solve the problem involve using the specific ratio of sides in a 45°-45°-90° triangle, which is 1:1:(sqrt{2}). Given the hypotenuse of 10:

The legs will be (frac{10}{sqrt{2}} 5sqrt{2}) units each.

Therefore, each leg is 5(sqrt{2}) units.

Alternatively, using the formula:

Length of one leg (frac{1}{sqrt{2}} times) hypotenuse

(frac{1}{sqrt{2}} times 10 5sqrt{2}) by rationalizing the denominator.

This confirms that each leg is 5(sqrt{2}) units.

Regardless of the method used, the legs of the 45°-45°-90° triangle with a hypotenuse of 13 units are approximately 9.213 units, 5(sqrt{2}) units (which is approximately 7.071 units), and 5(sqrt{2}) units (about 7.071 units).