Problem 71 - Algebra for College Students

Question

\(a\) and \(b\) represent the lengths of the legs of a right triangle, and \(c\) represents the length of the hypotenuse. Express answers in simplest radical form. Find \(c\) if \(a=4\) centimeters and \(b=6\) centimeters.

Step-by-Step Solution

Answer

The length of the hypotenuse \(c\) is \(2\sqrt{13}\) cm.

Step 1: Understand the problem

You are given a right triangle with two legs of lengths 4 cm and 6 cm and need to find the hypotenuse using the Pythagorean theorem.

Step 2: Apply the Pythagorean theorem

According to the Pythagorean theorem, the relationship between the lengths of the sides of a right triangle is given by: \(c^2 = a^2 + b^2\). In this case, substitute \(a = 4\) and \(b = 6\) into the equation.

Step 3: Calculate squares of sides

Compute \(4^2\) and \(6^2\). \(4^2 = 16\) and \(6^2 = 36\).

Step 4: Sum the squares

Add the squares of the legs: \(16 + 36 = 52\).

Step 5: Solve for \(c\)

Substitute the sum into the Pythagorean equation to find \(c^2\): \(c^2 = 52\).

Step 6: Take the square root

To find \(c\), take the square root of 52: \(c = \sqrt{52}\). Simplify \(\sqrt{52}\) into simplest radical form: \(c = 2\sqrt{13}\).

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This specific angle is called a right angle, and it gives the triangle some special properties. In a right triangle, the two shorter sides are called the legs, and they meet at the right angle. These legs are usually labeled as \(a\) and \(b\). The longest side, which is opposite the right angle, is known as the hypotenuse and is labeled as \(c\).

The special property of right triangles that makes them very important in geometry is the Pythagorean Theorem. This theorem provides a way to calculate the length of one of the sides if the lengths of the other two are known. This is extremely useful in various applications, including architecture and trigonometry. Always remember:

• The sum of squares of the two legs equals the square of the hypotenuse.
• This unique property holds true only in right triangles.

Hypotenuse

The hypotenuse is the longest side of a right triangle and is always opposite the right angle. It's crucial because it links directly to the Pythagorean Theorem, which helps us find its length based on the triangle's legs. To find the hypotenuse, you apply the theorem's formula: \(c^2 = a^2 + b^2\).

Consider a right triangle where the legs are known, like in the exercise with \(a = 4\) cm and \(b = 6\) cm. Use the equation:\[c^2 = a^2 + b^2\]

Calculate \(a^2 = 16\) and \(b^2 = 36\). Then, \(c^2 = 16 + 36 = 52\). Thus, the hypotenuse \(c\) can be found by taking the square root of 52. It's fascinating how this important property allows us to solve geometry problems easily.

Simplest Radical Form

The simplest radical form is a way of expressing square roots that are neat and organized. Simplifying radicals helps make mathematical expressions easier to work with and understand. In the context of the exercise, the hypotenuse \(c\) is expressed as \(\sqrt{52}\).

To express this in simplest radical form, you need to factor the number inside the square root into pairs of perfect squares. For example, \(52 = 4 \times 13\). The square root of 4 is 2, which can be taken out of the square root, resulting in: \(2\sqrt{13}\).

By doing this, you simplify \(\sqrt{52}\) into \(2\sqrt{13}\), which is much easier to work with and discuss. This practice of expressing in simplest radical form is especially important in exams and problems where clarity and precision in mathematical expressions are necessary. Remember, this approach helps in presenting the clearest possible answer, making complex concepts more digestible.