Problem 38 - Elementary Algebra Within Reach

Question

Geometry The longest side of a triangle is 3 times the length of the shortest side. The third side of the triangle is 4 inches longer than the shortest side. The perimeter is 49 inches. What are the lengths of the three sides of the triangle?

Step-by-Step Solution

Answer

The lengths of the three sides of the triangle are 9 inches, 27 inches, and 13 inches respectively.

Step 1: Define the Sides of the Triangle

Denote the shortest side of the triangle as \(s\). Then the longest side is \(3s\) and the third side is \(s+4\).

Step 2: Sum Up the Sides

The perimeter of the triangle is the sum of the sides. Therefore,\[49 = s + 3s + (s + 4)\]

Step 3: Solve for 's'

Solve the equation from Step 2 for \(s\), which is the shortest side. This gives you: \(49 = 5s + 4\), and upon rearrangement you have \(5s = 45\). Dividing through by 5 then gives \(s = 9\). Using the values of \(s\), calculate each side of the triangle. The shortest side \(s\) equals 9 inches, the longest side \(3s\) equals \(27\) inches, and the third side \(s+4\) equals \(13\) inches.

Key Concepts

Geometry Problems

Geometry problems often involve understanding and solving for shapes and their properties, such as lengths, areas, and angles. Here, the problem is about finding the sides of a triangle based on some given conditions. To tackle geometry problems, it's important to first read the problem carefully and identify what is being asked.
Next, define any unknowns, such as side lengths, with variables. This particular problem gives us relationships between the sides of a triangle, which we need to express mathematically:

• Shortest side: Let's call it \(s\).
• Longest side: 3 times the shortest side, or \(3s\).
• Third side: 4 inches longer than the shortest side, or \(s + 4\).

Making these relationships clear is crucial, as it sets the stage for solving the rest of the problem systematically.

Solving Equations

Solving equations is a fundamental skill in tackling geometry problems, as it allows us to find the values of unknowns by following logical steps. First, we use the information given about the triangle's perimeter to create an equation. The perimeter of a triangle is simply the sum of all its sides, which in this case is set to 49 inches:

\[49 = s + 3s + (s + 4)\]

Now simplify the expression by combining like terms:
- \(s\) + \(3s\) + \(s + 4\) becomes \(5s + 4\).

Rewriting the equation, we have:
\[49 = 5s + 4\]

To isolate \(s\), we start by subtracting 4 from both sides:
\[49 - 4 = 5s\]
Giving us:
\[45 = 5s\]

Finally, divide both sides by 5 to solve for \(s\):
\[s = 9\]

Now we know the shortest side of the triangle is 9 inches. Using what we've found, we can easily calculate the other sides as well.

Algebraic Expressions

Algebraic expressions allow us to model real-world relationships and constraints into mathematical forms. They involve numbers, variables, and operational symbols. In our exercise, we use algebraic expressions to represent the lengths of the triangle sides based on a single variable, \(s\). This form of expressions are valuable because they simplify complex relationships into manageable formulas.

For example:

• The longest side is expressed as \(3s\).
• The third side is expressed as \(s + 4\).
• The perimeter involves an expression \(s + 3s + (s + 4)\).

Breaking down the problem into these expressions helps in systematically solving it. Once \(s\) is known, substituting it back into each expression calculates the actual lengths. This structured use of algebraic expressions not only aids in understanding but also in ensuring accuracy.