Problem 43 - Algebra 2 and Trigonometry

Question

In \(43-47,\) express each answer in simplest radical form. The lengths of the sides of a triangle are \(\sqrt{75}\) inches, \(\sqrt{27}\) inches, and \(\sqrt{108}\) inches. What is the perimeter of the triangle?

Step-by-Step Solution

Answer

The perimeter is \( 14\sqrt{3} \) inches.

Step 1: Simplify each radical

Start by simplifying each of the side lengths. For \( \sqrt{75} \), rewrite it as \( \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \).For \( \sqrt{27} \), rewrite it as \( \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3} \).For \( \sqrt{108} \), rewrite it as \( \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3} \).

Step 2: Add the simplified radicals

Now add the simplified values of the side lengths to find the perimeter.\( 5\sqrt{3} + 3\sqrt{3} + 6\sqrt{3} = (5+3+6)\sqrt{3} = 14\sqrt{3} \).

Step 3: Present the perimeter

The perimeter of the triangle is \( 14\sqrt{3} \) inches, expressed in the simplest radical form.

Key Concepts

Understanding Triangle Perimeter

The perimeter of a triangle is simply the total distance around the triangle. To find it, you just add up the lengths of all three sides. In this exercise, we're dealing with unusual side lengths because they are given as square roots, or radicals, such as \( \sqrt{75} \), \( \sqrt{27} \), and \( \sqrt{108} \). But don't worry, they can be simplified, which is what we're going to do. Once simplified, we can easily add them together to find the perimeter.

When side lengths are provided as radicals, like in this problem, there's an extra step involved: simplifying these radicals to make addition easier. Understanding how to handle these radicals is key in these calculations. So, once you reduce each square root to its simplest form, adding them up becomes straightforward, just like adding plain numbers.

Simplifying Radicals

Simplifying radicals is like tidying up an expression. You make it as simple as it can be by reducing the factors inside the square root. Here's how you do it:

• First, factor the number inside the square root into a product of two numbers, with one ideally being a perfect square (like 4, 9, 16, 25).
• For \( \sqrt{75} \), you can break it down into \( 25 \times 3 \). The perfect square here is 25.
• This means \( \sqrt{75} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \), since \( \sqrt{25} = 5 \).

By repeating this for \( \sqrt{27} \) and \( \sqrt{108} \), you discover their simplest forms are \( 3\sqrt{3} \) and \( 6\sqrt{3} \) respectively. This process not only makes the numbers tidier, but also reveals a common factor, allowing for easier manipulation in further calculations.

Adding and Simplifying Algebraic Expressions Involving Radicals

Once you simplify each radical, the next step is to add these radicals together. This is required to find the triangle's perimeter when its sides are in simplest radical form. Luckily, adding radicals is quite similar to adding numbers: you just need like terms.

In our case, each side became something times \( \sqrt{3} \) \((5\sqrt{3}, 3\sqrt{3}, 6\sqrt{3})\). Since they all have a common factor of \( \sqrt{3} \), you can add their coefficients:

• Combine \( 5\sqrt{3} + 3\sqrt{3} + 6\sqrt{3} \).
• Add the numbers in front of the \( \sqrt{3} \): \( 5 + 3 + 6 \).
• This results in \( 14\sqrt{3} \).

So, the triangle's perimeter is expressed thoughtfully in simplest radical form as \( 14\sqrt{3} \). This simplifies the expression while preserving accuracy, making it easier to read and work with.