Question: The rectangle and triangle below have the same area.
Calculate $d$.
Calculator not allowed
8 cm
5 cm
5 cm
$d$
Not to scale
Rectangle (left, center-left): a purple rectangle with height 8 cm and width 5 cm.
Triangle (right, center-right): a yellow triangle with base $d$ and perpendicular height 5 cm.
1. **State the problem:** We have a rectangle and a triangle with the same area. The rectangle has height $8$ cm and width $5$ cm. The triangle has base $d$ cm and height $5$ cm. We need to find $d$.
2. **Formula for area:**
- Area of rectangle = height $\times$ width
- Area of triangle = $\frac{1}{2} \times$ base $\times$ height
3. **Calculate the area of the rectangle:**
$$\text{Area}_{\text{rectangle}} = 8 \times 5 = 40 \text{ cm}^2$$
4. **Set the areas equal:**
Since the areas are the same,
$$40 = \frac{1}{2} \times d \times 5$$
5. **Solve for $d$:**
Multiply both sides by 2 to eliminate the fraction:
$$2 \times 40 = \cancel{2} \times \frac{1}{\cancel{2}} \times d \times 5 \implies 80 = d \times 5$$
Divide both sides by 5:
$$\frac{80}{\cancel{5}} = d \times \frac{\cancel{5}}{5} \implies 16 = d$$
6. **Final answer:**
$$d = 16 \text{ cm}$$
This means the base of the triangle is $16$ cm to have the same area as the rectangle.