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.