Let us solve the problem step by step.

Problem Analysis:

1. Rectangle: The area of a rectangle is given by: $\text{Area} = \text{Length} \times \text{Width}.$ Here, the rectangle's dimensions are $8 \, \text{cm}$ (height) and $5 \, \text{cm}$ (width). Thus, the area of the rectangle is: $\text{Area}_{\text{rect}} = 8 \times 5 = 40 \, \text{cm}^2.$

2. Triangle: The area of a triangle is given by: $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}.$ Here, the triangle's height is $5 \, \text{cm}$, and the base is $d \, \text{cm}$. The area of the triangle is: $\text{Area}_{\text{tri}} = \frac{1}{2} \times d \times 5 = \frac{5d}{2} \, \text{cm}^2.$

3. Since the two areas are equal, we can set: $\text{Area}_{\text{rect}} = \text{Area}_{\text{tri}}.$

Solving the Equation:

Equating the two areas: $40 = \frac{5d}{2}.$

Multiply through by 2 to eliminate the fraction: $80 = 5d.$

Divide both sides by 5: $d = \frac{80}{5} = 16.$

Final Answer:

The value of $d$ is: $\boxed{16 \, \text{cm}}.$

Let me know if you want further clarifications!

Related Questions:

1. What is the general formula for calculating the area of a triangle?
2. How can you verify that two shapes have the same area mathematically?
3. If the base $d$ of the triangle was doubled, how would its area change?
4. What happens to the rectangle's area if one side is increased by 50%?
5. How do we calculate the areas of other polygons like trapeziums?

Tip:

Always include units in your calculations to ensure clarity and precision in math problems.