Problem 12 - Intermediate Algebra

Question

Solve for \(c,\) where \(c\) represents the length of the hypotenuse of a right triangle. Simplify the result, if possible. a. \(c^{2}=64\) b. \(c^{2}=15\) c. \(c^{2}=24\)

Step-by-Step Solution

Answer

a. \(c = 8\); b. \(c = \sqrt{15}\); c. \(c = 2\sqrt{6}\).

Step 1: Understanding the Problem

We are given that the squares of lengths denoted by \(c\), representing the hypotenuse of a right triangle, are provided in equations. We need to solve for \(c\) from these square equations.

Step 2: Solving Part a - c^2 = 64

Given \(c^2 = 64\), take the square root of both sides to solve for \(c\). So, \(c = \sqrt{64}\). Calculate \(\sqrt{64}\) to get \(c = 8\).

Step 3: Solving Part b - c^2 = 15

Given \(c^2 = 15\), take the square root of both sides. Thus, \(c = \sqrt{15}\). Since \(15\) is not a perfect square, the result for \(c\) remains irrational at \(\sqrt{15}\).

Step 4: Solving Part c - c^2 = 24

Given \(c^2 = 24\), take the square root of both sides. Therefore, \(c = \sqrt{24}\). Simplify \(\sqrt{24}\) by recognizing \(24 = 4 \times 6\), thus \(\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}\).

Key Concepts

Understanding Square Roots

The square root is an essential mathematical operation, especially when working with right triangles and their hypotenuse. In simpler terms, a square root finds the number that when multiplied by itself results in a given number. For example, the square root of 64 is 8, written as \(\sqrt{64} = 8\), because \(8 \times 8 = 64\).

• Square roots are denoted by the radical symbol \(\sqrt{}\).
• If a number is a perfect square, such as 64, its square root is an integer.
• Otherwise, the square root is expressed as an irrational number, as with \(\sqrt{15}\).

Taking the square root is crucial for finding hypotenuse lengths in right triangles, helping simplify equations like \(c^2 = 64\) to straightforward values like \(c = 8\). Understanding how to compute and simplify square roots helps solve geometric problems accurately.

Defining Right Triangles

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This unique property distinguishes right triangles from other triangles, playing a crucial role in geometry due to the Pythagorean Theorem. The sides of a right triangle hold special significance:

• The longest side opposite the right angle is called the hypotenuse.
• The other two sides are known as the legs.

The relationships in a right triangle help solve various mathematical problems, especially when using the formula \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse and \(a\) and \(b\) are the legs. Understanding these relationships is pivotal not only for calculating side lengths but also for applications in real-world scenarios, such as construction and navigation.

The Role of the Hypotenuse

In right triangles, the hypotenuse is the star player. It's the longest side and is always opposite the 90-degree angle. The hypotenuse holds significant value because of how it features in the Pythagorean Theorem, aiding in calculations for many triangular properties.

• Formulated as \(c = \sqrt{a^2 + b^2}\), it shows the relationship with the triangle's legs.
• For problems like \(c^2 = 64\), we solve for \(c\) to determine the hypotenuse length.
• The hypotenuse length helps ascertain distances and measures accurately.

Knowing how to determine the hypotenuse is especially useful in ensuring formulas and calculations are correct. Whether simplifying expressions where \(c^2 = 24\) becomes \(c = 2\sqrt{6}\), understanding this concept elevates both mathematical computations and everyday problem-solving efforts.