Question: The rectangle and triangle below have the same area.
Calculate $d$.
7 cm
5 cm
5 cm
$d$
Not to scale
Rectangle at bottom-left: a vertical rectangle with height $7$ cm and width $5$ cm.
Triangle at bottom-right: an isosceles-looking triangle with base $d$ along the bottom and a vertical height of $5$ cm from the apex to the base.
work out
1. **State the problem:** We have a rectangle and a triangle with the same area. The rectangle has height $7$ cm and width $5$ cm. The triangle has height $5$ cm and base $d$ 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:**
$$\text{Area}_{\text{rectangle}} = 7 \times 5 = 35 \text{ cm}^2$$
4. **Set the triangle area equal to rectangle area:**
$$\text{Area}_{\text{triangle}} = \text{Area}_{\text{rectangle}}$$
$$\frac{1}{2} \times d \times 5 = 35$$
5. **Solve for $d$:**
$$\frac{1}{2} \times 5 \times d = 35$$
$$\frac{5}{2} d = 35$$
Multiply both sides by $\cancel{\frac{2}{5}}$ to isolate $d$:
$$d = 35 \times \cancel{\frac{2}{5}}$$
$$d = 35 \times \frac{2}{5}$$
$$d = \frac{35 \times 2}{5} = \frac{70}{5} = 14$$
6. **Answer:**
$$d = 14 \text{ cm}$$
So, the base $d$ of the triangle is $14$ cm.