Problem 40 - Algebra 2 and Trigonometry

Question

The length of one leg of an isosceles right triangle is 6 inches. Express the length of the hypotenuse in simplest radical form.

Step-by-Step Solution

Answer

The hypotenuse is \(6\sqrt{2}\) inches long.

Step 1: Understand the Properties of an Isosceles Right Triangle

An isosceles right triangle has two equal legs with a right angle (90 degrees) formed where they meet. In this triangle, the legs are equal in length, and the hypotenuse is opposite the right angle.

Step 2: Apply the Pythagorean Theorem

The Pythagorean Theorem states that for a right triangle, \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the legs and \(c\) is the hypotenuse. For an isosceles right triangle, both legs are equal, so let their lengths be \(a = b = 6\).

Step 3: Substitute Values into the Pythagorean Theorem

Substitute \(a = 6\) and \(b = 6\) into the Pythagorean Theorem: \(6^2 + 6^2 = c^2\).

Step 4: Solve for the Hypotenuse

Calculate \(6^2 + 6^2\), which gives \(36 + 36 = 72\). Therefore, \(c^2 = 72\).

Step 5: Simplify the Expression for the Hypotenuse

Find \(c\) by taking the square root of 72. Simplifying \(c = \sqrt{72}\), note that \(72 = 36 \times 2 = 6^2 \times 2\). So, \(\sqrt{72} = \sqrt{6^2 \times 2} = 6\sqrt{2}\).

Key Concepts

Hypotenuse

In a right triangle, the hypotenuse is the side opposite the right angle. It is always the longest side of the triangle. If you have an isosceles right triangle, like in this exercise, the two legs are of equal length. This unique characteristic is crucial because it allows us to easily determine the hypotenuse using a simple ratio. If each leg has a length of 6 inches, as given, then you can find the hypotenuse using the properties of such a triangle and mathematical formulas.
To calculate the length of the hypotenuse, you don't need to reinvent the wheel. Instead, employ the Pythagorean Theorem, which we'll discuss in detail. The computation ultimately shows that in an isosceles right triangle, the hypotenuse is equal to the length of a leg multiplied by the square root of two. This conclusion comes from solving the Pythagorean equation, which results in the simplest radical form of the hypotenuse.

Pythagorean Theorem

The Pythagorean Theorem is an essential principle in geometry, particularly for right triangles. It states that in a right triangle, the sum of the squares of the two legs (\(a^2 + b^2\)) is equal to the square of the hypotenuse (\(c^2\)). In mathematical terms, this is written as:

• \(a^2 + b^2 = c^2\)

With this theorem, we can solve for the unknown side when the two other sides are known. In this exercise with an isosceles right triangle, both legs have the same length of 6 inches. This simplifies the process because you substitute both \(a = 6\) and \(b = 6\) into the equation, giving you:

• \(6^2 + 6^2 = c^2\)
• \(36 + 36 = c^2\)
• \(72 = c^2\)

Recognizing the symmetry of the problem can greatly simplify solving the theorem. As you see, calculating these values allows us to determine the length of the hypotenuse by solving for \(c\)\.

Simplest Radical Form

Simplifying expressions is a common requirement in math to make results more manageable and precise. The concept of simplest radical form comes into play when the result of an operation, like finding a square root, yields a non-perfect square. Here, when we solved for the hypotenuse \(c\), we found that \(c^2 = 72\).This means the hypotenuse \(c\) is equal to \(\sqrt{72}\).
To express \(\sqrt{72}\)in its simplest radical form, factor the number under the square root to its prime components:

• \(72 = 36 \times 2\)
• \(36 = 6^2\)

Using the properties of square roots, \(\sqrt{36 \times 2} = \sqrt{6^2 \times 2} = 6\sqrt{2}\).Now, the hypotenuse is expressed as \(6\sqrt{2}\). This is cleaner and often preferred because it shows the simplest form of the expression. Knowing how to simplify radicals is crucial for algebra and is widely used in geometry, physics, and engineering contexts.