To solve this problem, we need to determine the length of the rectangle. The key information provided is that the rectangle and the triangle have the same area.

Step 1: Find the Area of the Triangle

The formula for the area of a triangle is:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

From the image:

• Base of the triangle = 12 cm
• Height of the triangle = 16 cm

$\text{Area of the triangle} = \frac{1}{2} \times 12 \times 16 = 96 \text{ cm}^2$

Step 2: Set Up the Area Equation for the Rectangle

The area of a rectangle is calculated by:

$\text{Area} = \text{length} \times \text{width}$

From the image, the width of the rectangle is given as 7.5 cm. Let the length be $L$. Since the area of the rectangle is equal to the area of the triangle, we have:

$96 = L \times 7.5$

Step 3: Solve for $L$

$L = \frac{96}{7.5} = 12.8 \text{ cm}$

Conclusion

The length of the rectangle is 12.8 cm.

Would you like further details or explanations on any part of this solution? Here are some related questions to consider:

1. How do you find the area of different geometric shapes?
2. Can the area formula for triangles be used for any triangle type?
3. What are some real-life applications of calculating the area?
4. How do you convert different units when calculating area?
5. How would the problem change if the triangle were a right triangle with different dimensions?

Tip: Always double-check your measurements and units when calculating areas, especially when comparing different shapes.