Problem 217 - Elementary Algebra

Question

In the following exercises, solve using triangle properties. If a triangle has sides 6 feet and 9 feet and the perimeter is 23 feet, how long is the third side?

Step-by-Step Solution

Answer

The third side is 8 feet.

Step 1: Recall the Perimeter Formula

The perimeter of a triangle is the sum of the lengths of its sides. If the sides of the triangle are denoted as a, b, and c, the perimeter P can be expressed as: \[ P = a + b + c \]

Step 2: Substitute Known Values

We are given two sides of the triangle, which are 6 feet and 9 feet, and the perimeter, which is 23 feet. Let the length of the third side be denoted by x. Thus, we substitute these values into the perimeter formula: \[ 6 + 9 + x = 23 \]

Step 3: Solve for the Unknown Side

To find the value of x, solve the equation: \[ 6 + 9 + x = 23 \] First, combine the known lengths: \[ 15 + x = 23 \] Next, isolate the variable x by subtracting 15 from both sides of the equation: \[ x = 23 - 15 \] Thus, \[ x = 8 \]

Key Concepts

Perimeter Formula

One of the fundamental concepts in geometry is the perimeter of a shape. For a triangle, the perimeter is the total distance around the triangle. This is calculated by adding up the lengths of all three sides.
For example, if a triangle has sides of lengths a, b, and c, the perimeter P is given by the formula: \[ P = a + b + c \]
Knowing this formula is extremely useful, as it allows you to understand how the sides of a triangle relate to its overall shape and size.

Solving for Unknowns

In many triangle problems, you may need to find the length of an unknown side. This involves substituting known values into the perimeter formula and then solving the resulting equation.
Let's go through the steps with the given triangle:

• The sides are 6 feet, 9 feet, and x feet (the unknown side).
• The perimeter is given as 23 feet.
• Using the perimeter formula, we have: \[ 6 + 9 + x = 23 \]
  Combining the known lengths simplifies the equation to:
  \( 15 + x = 23 \)
  To isolate x, subtract 15 from both sides:
  \( x = 23 - 15 \)
  Solving this, we get:
  \( x = 8 \)

This shows how you can find the unknown side by using basic algebra.

Basic Algebra

Basic algebra is key in solving many geometry problems, especially when an equation involves unknown variables. Let's break down the steps used in our problem:

• First, set up your equation using the known values. In our case: \[ 6 + 9 + x = 23 \]
• Then combine the constants on one side of the equation: \[ 15 + x = 23 \]
• Finally, isolate the variable (x) by performing inverse operations. Here, we subtracted 15 from both sides to get: \[ x = 23 - 15 = 8 \]

Through these steps, algebra helps us solve for unknowns, making it easier to handle various mathematical problems.
Understanding these concepts equips you with the tools to tackle numerous mathematical and real-world scenarios involving triangles.