Problem 12 - Elementary and 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 need to find the length of the hypotenuse, denoted by \(c\), of a right triangle given the equations \(c^2 = 64\), \(c^2 = 15\), and \(c^2 = 24\). This is typically solved by taking the square root of both sides of the equation.

Step 2: Solving Part (a)

For \(c^2 = 64\), take the square root of both sides to get \(c = \sqrt{64}\). Calculate \(\sqrt{64}\), which equals 8. Therefore, \(c = 8\).

Step 3: Solving Part (b)

For \(c^2 = 15\), take the square root of both sides to get \(c = \sqrt{15}\). Since 15 is not a perfect square, \(\sqrt{15}\) is already in its simplest form.

Step 4: Solving Part (c)

For \(c^2 = 24\), take the square root of both sides to get \(c = \sqrt{24}\). Simplify \(\sqrt{24}\) by factoring inside the square root: \(\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}\). Thus, \(c = 2\sqrt{6}\).

Key Concepts

Right Triangle

A right triangle is a special type of triangle that includes one angle measuring exactly 90 degrees. This angle is known as the right angle, and it creates a distinctive relationship between the triangle's sides, known as the Pythagorean theorem. The sides adjacent to the right angle are called the "legs" of the triangle, whereas the side opposite the right angle is called the "hypotenuse." The hypotenuse is always the longest side of a right triangle.

Understanding the properties of right triangles can help solve problems involving their side lengths. When given one or two side lengths, you can use the Pythagorean theorem to find the missing length. This theorem is represented by the equation: \[ a^2 + b^2 = c^2 \] where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse. Knowing how to work with right triangles is essential for solving many geometric problems.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted with the radical symbol \(\sqrt{}\). Finding the square root is the inverse operation of squaring a number.

When solving for the hypotenuse in problems involving right triangles, you often take the square root of an expression. For example, if you are given that the hypotenuse squared equals 64 (\(c^2 = 64\)), you find \(c\) by calculating \(\sqrt{64}\). This simplifies to 8, since \(8 \times 8 = 64\).

When numbers are not perfect squares, such as 15, their square roots can be approximated or left in a radical form (e.g., \(\sqrt{15}\) remains \(\sqrt{15}\)). However, if possible, simplify expressions under the square root by factoring out perfect squares, as seen in the case of \(\sqrt{24}\), which simplifies to \(2\sqrt{6}\).

• Square roots are crucial for solving quadratic equations.
• Simplifying square root expressions can make calculations easier.

Hypotenuse

The hypotenuse is the side opposite the right angle in a right triangle and is the longest side. Its length is determined using the Pythagorean theorem, particularly useful when solving exercises like finding the hypotenuse from given values such as \(c^2 = 64\), \(c^2 = 15\), or \(c^2 = 24\).

To find the hypotenuse length \(c\), we typically solve \(c^2\) equations by taking the square root, as shown step-by-step:- For \(c^2 = 64\), compute \(c = \sqrt{64} = 8\).- For \(c^2 = 15\), \(c\) is the simplified \(\sqrt{15}\), as it isn't a perfect square.- For \(c^2 = 24\), simplify \(c = \sqrt{24}\) to \(2\sqrt{6}\).

Knowing how to determine the hypotenuse helps in a wide array of applications, from simple geometric constructions to more complex architectural designs.