Problem 44 - Algebra 2 and Trigonometry

Question

In \(43-47,\) express each answer in simplest radical form. The length of each of the two congruent sides of an isosceles triangle is \(\sqrt{500}\) feet and the length of the third side is \(\sqrt{45}\) feet. What is the perimeter of the triangle?

Step-by-Step Solution

Answer

The perimeter of the triangle is \( 23\sqrt{5} \) feet.

Step 1: Understand the Definition of Perimeter

The perimeter of a triangle is the sum of the lengths of all its sides. In this case, we have an isosceles triangle with two congruent sides and a base.

Step 2: Simplify the Radical for Congruent Sides

The length of each congruent side is given as \( \sqrt{500} \). Simplify \( \sqrt{500} \) by finding its prime factors: \( 500 = 5^3 \times 2^2 \). So, \( \sqrt{500} = \sqrt{(5^2 \times 5) \times (2^2)} = \sqrt{5^2 \times 2^2 \times 5} = 10\sqrt{5} \).

Step 3: Simplify the Radical for the Third Side

The length of the third side is \( \sqrt{45} \). Simplify \( \sqrt{45} \) by finding its prime factors: \( 45 = 3^2 \times 5 \). Therefore, \( \sqrt{45} = \sqrt{3^2 \times 5} = 3\sqrt{5} \).

Step 4: Add the Simplified Side Lengths

Now that we have simplified the side lengths, add them together to find the perimeter: \( 10\sqrt{5} + 10\sqrt{5} + 3\sqrt{5} \).

Step 5: Combine Like Terms

Since the terms are like terms, combine them: \( (10+10+3)\sqrt{5} = 23\sqrt{5} \).

Step 6: Express the Answer in Simplest Radical Form

Thus, the perimeter of the triangle, expressed in simplest radical form, is \( 23\sqrt{5} \) feet.

Key Concepts

Simplifying Radicals

Understanding how to simplify radicals is like untangling a complex knot into a neat string. When you see a radical expression, like \(\sqrt{500}\) or \(\sqrt{45}\), the goal is to simplify it into a form that is much easier to understand and use. Here's how:

• First, find the prime factors of the number under the square root.
• Pair up the same numbers in these prime factors because pairs can be simplified to a single integer outside the root.
• Any numbers left unpaired stay inside the square root.

For example, for \(\sqrt{500}\), its prime factors are \(5^3 \times 2^2\). You take one 5 and one 2 out of the root as a pair, getting \(10\sqrt{5}\). For \(\sqrt{45}\), we have \(3^2 \times 5\), which simplifies to \(3\sqrt{5}\) after taking 3 out of the root.

Congruent Sides

In an isosceles triangle, two sides are always congruent, meaning they are of equal length. This special feature is crucial when calculating the perimeter. The congruent sides will always be the same length, which makes them fairly straightforward to work with in problems.

• Being congruent simply means they mirror each other in length.
• In our exercise, these sides are given as \(\sqrt{500}\), which after simplification are each \(10\sqrt{5}\).

Recognizing congruency is important as it can simplify calculations and help to solve problems more efficiently.

Triangle Perimeter Calculation

Calculating the perimeter of a triangle is a fundamental math concept. It's straightforward for any triangle — simply add up the lengths of all three sides. In the case of an isosceles triangle, knowing that two sides are congruent further simplifies the operation.

• Sum the length of both congruent sides and the base (the third side).
• In our triangle, after simplification, the sides measure as \(10\sqrt{5}, 10\sqrt{5}, \) and \(3\sqrt{5}\).

Adding these gives the equation for the perimeter as \(10\sqrt{5} + 10\sqrt{5} + 3\sqrt{5}\), which simplifies to \(23\sqrt{5}\). Once you understand this, calculating for any isosceles triangle becomes a breeze.

Radical Expressions

Radical expressions involve roots, mostly square roots, and can initially look intimidating. But simplifying them transforms complexity into clarity. Let's break it down:

• Square roots like \(\sqrt{500}\) or \(\sqrt{45}\) are examples of radical expressions.
• Simplifying involves converting these into their simplest form by reducing them with methods like finding prime factors.
• Ultimately, simplified radical expressions often become numbers with roots, giving them a cleaner, more usable form, such as \(10\sqrt{5}\) or \(3\sqrt{5}\).

These simpler forms are easier to use and understand in further calculations, such as adding to find a perimeter.