Tina is designing a square shaped tile with a border. the area of the tile is 64 square centimeters. if she increases the side length of the pool by 2cm, ghe new area is 100 square centimeters . what was the original side length of the tile? 

To solve this problem, we need to find the original side length of the square-shaped tile. Let's work through the problem step by step.

### Step 1: Understanding the Problem
1. The area of the original tile is given as \( 64 \, \text{cm}^2 \).
2. The side length of the square tile increases by \( 2 \, \text{cm} \), making the new area \( 100 \, \text{cm}^2 \).

### Step 2: Setting Up the Equations
Let the original side length of the tile be \( x \) cm.

1. **Original Area:**
   \[
   \text{Area} = x^2 = 64 \, \text{cm}^2
   \]
   Solving for \( x \):
   \[
   x = \sqrt{64} = 8 \, \text{cm}
   \]

So, the original side length of the tile is \( 8 \, \text{cm} \).

2. **New Side Length After Increase:**
   After increasing the side length by \( 2 \, \text{cm} \), the new side length becomes \( x + 2 \) cm.

   Therefore, the new side length is:
   \[
   8 + 2 = 10 \, \text{cm}
   \]

3. **New Area:**
   The new area with the increased side length is:
   \[
   \text{New Area} = 10^2 = 100 \, \text{cm}^2
   \]

### Conclusion
The original side length of the tile was \( 8 \, \text{cm} \).

Would you like further details or have any other questions related to this problem? Here are some additional questions you might explore:

1. What is the perimeter of the original tile?
2. How does the perimeter change after the side length is increased by 2 cm?
3. If Tina added a border around the tile, what would be the new dimensions of the tile with the border?
4. How would the area change if Tina increased the side length by 3 cm instead of 2 cm?
5. What is the ratio of the new area to the original area after increasing the side length by 2 cm?

**Tip:** When solving geometry problems involving squares, remember that the area is always the side length squared, which can help you easily find side lengths from areas.

Learn how to find the original side length of a square tile when its area increases from 64 square centimeters to 100 square centimeters after increasing the side length by 2 cm. This problem involves basic geometry concepts and is suitable for students in Grades 7-9.

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