Solve the problem below by defining a variable and then writing and solving an equation. Homework Help ✎ The perimeter of a triangle is 31 cm. Sides #1 and #2 have equal length, while Side #3 is one centimeter shorter

than twice the length of Side #1. How long is each side?

To solve this problem, let's define a variable and set up an equation.

### Step 1: Define the variable
Let:
- \( x \) = the length of Side #1 (in cm).

Since Side #1 and Side #2 are of equal length, the length of Side #2 is also \( x \).

Side #3 is one centimeter shorter than twice the length of Side #1. Therefore, the length of Side #3 is \( 2x - 1 \) cm.

### Step 2: Set up the equation
The perimeter of the triangle is the sum of the lengths of all three sides. According to the problem, the perimeter is 31 cm. Therefore, we can write the equation as:
\[
x + x + (2x - 1) = 31
\]

### Step 3: Simplify and solve the equation
Simplify the equation:
\[
2x + 2x - 1 = 31
\]
Combine like terms:
\[
4x - 1 = 31
\]

Add 1 to both sides to isolate the term with \( x \):
\[
4x = 32
\]

Finally, divide both sides by 4 to solve for \( x \):
\[
x = 8
\]

### Step 4: Find the length of each side
- Side #1: \( x = 8 \) cm
- Side #2: \( x = 8 \) cm
- Side #3: \( 2x - 1 = 2(8) - 1 = 16 - 1 = 15 \) cm

### Final Answer:
The lengths of the sides of the triangle are:
- Side #1 = 8 cm
- Side #2 = 8 cm
- Side #3 = 15 cm

Would you like more details on any of the steps? Here are some related questions:

1. How do you find the perimeter of a triangle?
2. What are some properties of isosceles triangles?
3. How do you solve equations with multiple variables?
4. What does it mean for a side to be "one centimeter shorter"?
5. Can you solve a similar problem with different side relationships?

**Tip:** When solving word problems, it's helpful to define a variable and set up an equation based on the relationships described in the problem. This approach makes it easier to find a solution.

Learn how to solve a triangle perimeter problem where two sides are equal and the third side is one centimeter shorter than twice the length of one of the equal sides. Detailed step-by-step solution provided. Suitable for Grades 7-9.

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