Problem 41 - Algebra 2 and Trigonometry

Question

The length of the hypotenuse of a right triangle is 24 meters and the length of one leg is 12 meters. Express the length of the other leg in simplest radical form.

Step-by-Step Solution

Answer

The length of the other leg is \(12\sqrt{3}\) meters.

Step 1: Identify the Right Triangle Components

In this problem, you are given a right triangle where the hypotenuse (the longest side) is 24 meters and one of the legs (let's call it \(a\)) is 12 meters. You need to find the length of the other leg (let's call it \(b\)).

Step 2: Apply the Pythagorean Theorem

To solve for the unknown leg \(b\), use the Pythagorean theorem: \(c^2 = a^2 + b^2\). Here, \(c = 24\) meters and \(a = 12\) meters. So, plug in the known values: \(24^2 = 12^2 + b^2\).

Step 3: Compute the Squares of Known Values

Calculate \(24^2\) and \(12^2\). You get \(24^2 = 576\) and \(12^2 = 144\). Substitute these into the equation: \(576 = 144 + b^2\).

Step 4: Solve for the Unknown Leg

Rearrange the equation to solve for \(b^2\): \(b^2 = 576 - 144\). Compute the subtraction: \(b^2 = 432\).

Step 5: Simplify the Radical Expression

To find \(b\), take the square root of both sides: \(b = \sqrt{432}\). Simplify \(\sqrt{432}\) by finding the prime factors of 432: 432 = 2^4 \times 3^3. Simplify \(\sqrt{432}\) to its simplest radical form: \(\sqrt{2^4 \times 3^3} = 2^2 \times 3 \times \sqrt{3} = 12\sqrt{3}\). Thus, the length of the other leg is \(12\sqrt{3}\) meters.

Key Concepts

Understanding Right Triangles

A right triangle is a special type of triangle that includes one angle measuring exactly 90 degrees. This right angle is what defines a right triangle. The right angle divides the triangle into two distinct sides: the two legs and the hypotenuse. Here, the hypotenuse is the side opposite the right angle and always the longest side in a right triangle. The other two sides are called the legs, and they form the two edges of the triangle that meet at the right angle. Knowing these definitions will help when applying principles like the Pythagorean theorem to solve problems.

The Hypotenuse and Its Importance

In a right triangle, the hypotenuse is crucial as it is directly involved in the Pythagorean theorem. This longest side is opposite the right angle and can be used along with one leg to find the other. In problems where the lengths of two sides are given, the hypotenuse's role helps determine the missing length, ensuring that all sides satisfy the equation of the Pythagorean theorem. Always remember that without the hypotenuse, the sides of a right triangle do not follow this equation.

The Role of Square Roots in Solving Right Triangles

Square roots are vital in solving for missing sides in right triangles using the Pythagorean theorem. When you apply the theorem, you often need to take the square root to solve for the length of a missing side. For instance, if you derive an equation like \(b^2 = 432\), you must take the square root of both sides to find \(b = \sqrt{432}\). This is important because it allows you to express the lengths of triangle sides in terms that can be simplified further.

Expressing in Simplest Radical Form

To present a derived square root in its simplest form, it's essential to break down the number into its prime factors. In this example, for \(\sqrt{432}\), you factor 432 into \(2^4 \times 3^3\). From there, simplify by taking out pairs of factors: two pairs of 2's and one pair of 3's can be taken outside the radical, which results in \(2^2 \times 3 \times \sqrt{3}\). Simplifying further gives you \(12\sqrt{3}\), which is much neater and shows the precise relationship between components. This approach can make working with radicals easier and more intuitive.