Let’s analyze the problem.

The problem describes an isosceles triangle with:

• Two equal sides.
• One side that is five times the length of the shorter side.
• A perimeter of at least 60 cm.

Let the length of the shorter side be $x$. Then:

• Two equal sides each have length $x$,
• The third side is $5x$,
• The perimeter is $2x + 5x = 7x$.

Step 1: Solve for $x$

The perimeter condition says: $7x \geq 60$ $x \geq \frac{60}{7} \approx 8.57 \, \text{cm}.$

Thus, the minimum integer value of $x$ is $9$ cm.

Step 2: Verify the side lengths

• The shorter sides: $x = 9 \, \text{cm}$,
• The longer side: $5x = 45 \, \text{cm}$.

The sides $9, 9, 45$ satisfy the triangle inequality rule:

1. $9 + 9 > 45$ is not true (the sides cannot form a triangle).

To satisfy the inequality, we need to increase $x$.

Step 3: Adjust $x$ for triangle inequality

For the inequality $2x > 5x$, we simplify: $x > \frac{45}{2} = 22.5 \, \text{cm}.$

So, the minimum value of $x$ is $23 \, \text{cm}$.

Final Answer:

• The shorter sides are $23 \, \text{cm}$,
• The longer side is $5(23) = 115 \, \text{cm}$.

The minimum side lengths are: $23, 23, 115 \, \text{cm}.$

Let me know if you’d like further explanation or details! Here are 5 related questions:

1. How does the triangle inequality rule work?
2. Can we solve similar problems for different perimeters?
3. What is the general approach to solving isosceles triangle problems?
4. How can this problem be represented graphically?
5. What are other real-world applications of the triangle inequality?

Tip: Always verify all constraints, especially the triangle inequality, when solving triangle-related problems!