Question: 9. The base and the height of a triangle are in the ratio $7:2$ and its area are $630$ cm$^2$. Find the length and the width of the rectangle.
1. **State the problem:** We are given a triangle with base and height in the ratio $7:2$ and area $630$ cm$^2$. We need to find the base and height of the triangle.
2. **Formula for the area of a triangle:**
$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$
3. **Express base and height in terms of a variable:**
Let the common ratio factor be $x$.
Then base $= 7x$ and height $= 2x$.
4. **Write the area equation:**
$$630 = \frac{1}{2} \times 7x \times 2x$$
5. **Simplify the equation:**
$$630 = \frac{1}{2} \times 14x^2 = 7x^2$$
6. **Solve for $x^2$:**
$$x^2 = \frac{630}{7} = 90$$
7. **Find $x$:**
$$x = \sqrt{90} = 3\sqrt{10}$$
8. **Calculate base and height:**
Base $= 7x = 7 \times 3\sqrt{10} = 21\sqrt{10}$ cm
Height $= 2x = 2 \times 3\sqrt{10} = 6\sqrt{10}$ cm
**Final answer:**
Base $= 21\sqrt{10}$ cm and Height $= 6\sqrt{10}$ cm.