Problem 230 - Elementary Algebra

Question

In the following exercises, solve using triangle properties. One side of a triangle is three times the shortest side. The third side is three feet more than the shortest side. The perimeter is 13 feet. Find the lengths of all three sides.

Step-by-Step Solution

Answer

The lengths of the sides are 2 feet, 6 feet, and 5 feet.

Step 1 - Define Variables

Let the shortest side be represented as \( x \) feet.

Step 2 - Express the Other Sides in Terms of x

One side is three times the shortest side, so it can be represented as \( 3x \) feet. The third side is three feet more than the shortest side, so it can be represented as \( x + 3 \) feet.

Step 3 - Set Up the Perimeter Equation

The perimeter of the triangle is the sum of all its sides. Therefore, the equation is: \[ x + 3x + (x + 3) = 13 \]

Step 4 - Simplify the Equation

Combine like terms to simplify the equation: \[ 5x + 3 = 13 \]

Step 5 - Solve for x

Subtract 3 from both sides to isolate the term with \( x \): \[ 5x = 10 \]Then, divide by 5: \[ x = 2 \]

Step 6 - Calculate the Lengths of the Other Sides

Substitute \( x = 2 \) back into the expressions for the other sides. The second side is \( 3x = 3 \times 2 = 6 \) feet. The third side is \( x + 3 = 2 + 3 = 5 \) feet.

Step 7 - Verify the Solution

Check that the perimeter equals 13 feet: \[ 2 + 6 + 5 = 13 \]Since this is correct, the lengths of the sides are verified.

Key Concepts

Solving Triangle Problems

When solving triangle problems, it's crucial to understand the properties and relationships between the sides and angles of triangles. Triangles have three sides and three angles, and there are established properties that help us solve problems involving them.

For example:
- The sum of the interior angles is always 180 degrees.
- The length of any side of a triangle is always less than the sum but greater than the difference of the other two sides.

In this example, we use the relationship between the sides and the perimeter to set up and solve an algebraic equation.
By defining variables and expressing the sides in terms of those variables, the problem simplifies into solving a simple algebra equation. Understanding these steps is essential in mastering triangle problems.

Perimeter of a Triangle

The perimeter of a triangle is the total distance around the triangle, which is found by adding the lengths of its three sides together.
For the given triangle problem, we started by letting the shortest side be represented as \( x \) feet.

Here is how we expressed each side:
- Shortest side = \( x \) feet.
- Another side = \( 3x \) feet (three times the shortest side).
- Third side = \( x + 3 \) feet (three feet more than the shortest side).

Then, we set up the perimeter equation:
\[ x + 3x + (x + 3) = 13 \]
By solving this equation, we confirmed that the perimeter is precisely 13 feet, ensuring our solution is correct and verifying the lengths of each side.

Algebraic Equations

Algebraic equations are fundamental tools in solving geometry problems, like finding the sides of a triangle given its perimeter.
Algebra allows us to create mathematical models to represent real-world problems.

In our example problem, defining the sides of the triangle helps set up the algebraic equation for the perimeter. We have:
- Shortest side: \( x \) feet.
- Another side: \( 3x \) feet.
- Third side: \( x + 3 \) feet.

We then combined these into a single equation:
\[ x + 3x + (x + 3) = 13 \]

Upon simplifying:
\[ 5x + 3 = 13 \]
We solved for \( x \) by isolating it:
\[ 5x = 10 \]
\[ x = 2 \]

Finally, substituting \( x = 2 \) back into the expressions for each side confirmed the lengths and verified our solution. This step-by-step process highlights how algebraic equations are crucial in breaking down and solving geometric problems.