Let the sides of the triangle be represented as $a$, $b$, and $c$, where:

• $a$ is the shortest side,
• $b$ is the middle-length side,
• $c$ is the longest side.

Step 1: Formulate the equations based on the given information.

1. The perimeter of the triangle is the sum of its three sides: $a + b + c = 71$

2. The longest side $c$ is 3 cm less than the sum of the other two sides: $c = a + b - 3$

3. Twice the shortest side $a$ is 8 cm less than the longest side: $2a = c - 8$

Step 2: Solve the system of equations.

First, solve for $c$ from equation (3): $2a = c - 8 \quad \Rightarrow \quad c = 2a + 8$

Now, substitute this expression for $c$ into equation (2): $2a + 8 = a + b - 3$ Simplify and solve for $b$: $2a + 8 = a + b - 3 \quad \Rightarrow \quad 2a - a + 8 + 3 = b \quad \Rightarrow \quad b = a + 11$

Next, substitute $c = 2a + 8$ and $b = a + 11$ into the perimeter equation $a + b + c = 71$: $a + (a + 11) + (2a + 8) = 71$ Simplify: $a + a + 11 + 2a + 8 = 71 \quad \Rightarrow \quad 4a + 19 = 71$ Solve for $a$: $4a = 71 - 19 \quad \Rightarrow \quad 4a = 52 \quad \Rightarrow \quad a = 13$

Step 3: Find the values of $b$ and $c$.

Now that $a = 13$, substitute it back into the expressions for $b$ and $c$:

$b = a + 11 = 13 + 11 = 24$ $c = 2a + 8 = 2(13) + 8 = 26 + 8 = 34$

Final Answer:

The lengths of the sides of the triangle are:

• $a = 13$ cm,
• $b = 24$ cm,
• $c = 34$ cm.

Would you like more details or have any questions about this solution?

Here are 5 related questions to further explore:

1. How would the solution change if the perimeter was 80 cm?
2. Can a triangle have sides in a ratio of 13:24:34?
3. What is the area of this triangle using Heron's formula?
4. How do you classify this triangle based on its side lengths?
5. What would happen if the longest side were equal to the sum of the other two sides?

Tip: When solving problems with multiple variables, it's helpful to simplify equations step-by-step before solving the entire system.