Problem 38 - Algebra A Combined Function

Question

The perimeter of a yield sign in the shape of an isosceles triangle is 22 feet. If the shortest side is 2 feet less than the other two sides, find the length of the shortest side.

Step-by-Step Solution

Answer

The shortest side is 6 feet long.

Step 1: Understand the Problem

We have an isosceles triangle where two sides are equal, and the shortest side is 2 feet less than the other two equal sides. The perimeter is 22 feet.

Step 2: Define Variables

Let's set the length of one of the equal sides of the triangle as \( x \). Thus, since the shortest side is 2 feet less than these, we can represent the shortest side as \( x - 2 \).

Step 3: Set Up the Equation

The perimeter of a triangle is the sum of the lengths of all its sides. Here, it is given as 22 feet. So, we can write the equation as:\[ x + x + (x - 2) = 22 \]

Step 4: Simplify the Equation

Combine like terms in the equation:\[ 3x - 2 = 22 \]

Step 5: Solve for the Variable

Add 2 to both sides to isolate the term with the variable:\[ 3x = 24 \]Next, divide by 3:\[ x = 8 \]

Step 6: Find the Shortest Side

The shortest side is 2 feet less than \( x \), so substitute \( x = 8 \) into \( x - 2 \):\[ 8 - 2 = 6 \]

Key Concepts

Triangle Perimeter Calculation

Calculating the perimeter of a triangle involves summing the lengths of all its sides. For a general triangle, you need to know or find the measurements of each side.
For example, if a triangle has sides with lengths of \( a \), \( b \), and \( c \), the perimeter \( P \) is simply \( P = a + b + c \).
This is applicable to any triangle, regardless of its type.

• For equilateral triangles, all sides are the same length, so the perimeter is \( 3 \times a \), where \( a \) is the length of a side.
• In isosceles triangles, like the one in our exercise, two sides are equal. Thus, its perimeter can be calculated with the formula \( x + x + (x - 2) \) if the shortest side is 2 feet less than the other two.
• For scalene triangles, you'll often have to measure each side separately before summing them up to find the perimeter.

Remember, the perimeter is always expressed in units of linear measurement, such as feet, meters, or inches.

Isosceles Triangle Properties

Isosceles triangles are unique in geometry because they have not just one, but two equal sides. This symmetry brings about several interesting properties:

• Equal Sides and Angles: The angles opposite these equal sides are also the same. This property is useful when solving for angles in geometric problems.
• Height and Base: In isosceles triangles, the height can be drawn from the vertex angle to the midpoint of the base. This forms two congruent right triangles, which is often used in trigonometry and calculations involving height.
• Perimeter Relation: When calculating the perimeter, knowing that two sides are equal simplifies the calculation, as seen in our example problem where you only need to solve for two variables instead of three.

These attributes make isosceles triangles easier to work with in many instances, as they provide symmetry that simplifies calculations and predictions.

Solving Algebraic Equations

Algebraic equations are mathematical statements that use variables to represent numbers or expressions. When solving them, the goal is to find the value of the variable that makes the equation true. Here’s how you tackle such problems:

• Define the Variables: First, understand what each symbol in your equation represents. In the exercise, \( x \) was used for the length of each of the equal sides.
• Set Up the Equation: Use the problem's conditions to form an equation. With the isosceles triangle problem, it was \( x + x + (x - 2) = 22 \).
• Isolate the Variable: Simplify the equation to get the variable by itself. Combine like terms and use operations such as addition, subtraction, multiplication, or division to solve for \( x \).
• Verify Your Solution: Once you find a solution, substitute it back into the original equation to verify it works. This ensures your answer satisfies the conditions given.

Solving algebraic equations is a fundamental skill in math, helping solve problems ranging from simple arithmetic to complex calculus.