Problem 24 - Algebra 1: Concepts and Skills

Question

Given a triangle whose sides have lengths \(a, b,\) and \(c,\) the formula for its perimeter \(P\) is \(P=a+b+c.\) The steps at the right show a formula for finding a side length of a triangle given the perimeter and the lengths of the two other sides. Write an explanation for each step.

Step-by-Step Solution

Answer

Given a triangle with sides of length 'a', 'b' and 'c' and a total perimeter of 'P', the length of 'a' can be isolated and calculated using the rearranged formula a = P - b - c.

Step 1: Understand the Perimeter Formula

The first step is realizing what the equation P = a + b + c represents. It is used in calculating the perimeter of a triangle. 'P' represents the total length around the triangle, while 'a, b, c' represent the lengths of the triangle's sides.

Step 2: Isolating Side 'a' in the Formula

To find the length of one side given the perimeter and the lengths of the two other sides, the given equation needs to be rearranged. This can be done using subtraction. If we want to solve for 'a', we isolate 'a' on one side by subtracting 'b' and 'c' from both sides. This gives us a = P - b - c.

Step 3: Explanation of the Isolated Formula

The final formula a = P - b - c can be read as: the length of side 'a' is equal to the total perimeter 'P' minus the length of the other two sides 'b' and 'c'. This effectively means that if you know the total length around the triangle (the perimeter) and the lengths of two of its sides, you can find out the length of the third side by subtraction.

Key Concepts

Triangle Sides

In geometry, a triangle is a three-sided polygon. Understanding the parts of a triangle is essential to solving problems related to its perimeter. A triangle has three sides, usually denoted as \(a\), \(b\), and \(c\). These sides can vary in length, leading to different types of triangles such as equilateral, isosceles, and scalene.

• An equilateral triangle has all three sides of equal length.
• An isosceles triangle has two sides of the same length.
• A scalene triangle has all three sides of different lengths.

To solve problems involving triangles, you need to know the lengths of the sides involved. When one side is unknown, other known properties of the triangle, like its perimeter, may help you determine it. Knowing each side's length is crucial for finding the perimeter.

Perimeter Formula

The perimeter of a triangle is the total distance around it. Calculating the perimeter is straightforward if you know the lengths of all three sides. The formula to find a triangle's perimeter is: \[P = a + b + c\]Where \(P\) is the perimeter, and \(a\), \(b\), and \(c\) are the lengths of the triangle's sides.

• This formula reveals how each side contributes to the triangle's total perimeter.
• If one side's length is changed, the total perimeter changes accordingly.

Understanding this formula is the foundation for solving problems where the perimeter is given, and one of the side lengths needs to be determined.

Solving for a Side of a Triangle

In some situations, you have the perimeter of a triangle and need to find the length of one side. If you know the perimeter \(P\) and the lengths of two sides, \(b\) and \(c\), but not \(a\), you can rearrange the perimeter formula to solve for \(a\).
To find \(a\), isolate it by subtracting the other two sides from the perimeter:\[a = P - b - c\]This formula implies:

• First, take the total perimeter \(P\).
• Subtract the sum of the known sides \(b\) and \(c\).
• The result gives you the length of the third side \(a\).

This step-by-step rearrangement of the formula helps you identify the missing side length based on the information given. It's a practical method used in geometry to find unknown quantities when dealing with triangles.