Gradient Parallelogram Beb667

1. **State the problem:** We need to find the gradients (slopes) of sides $AB$ and $CD$ of the parallelogram $ABCD$ with points $A(2,2)$, $B(8,6)$, $C(6,8)$, and $D(0,4)$. 2. **Formula for gradient:** The gradient $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 3. **Calculate gradient of $AB$:** Points: $A(2,2)$ and $B(8,6)$ $$m_{AB} = \frac{6 - 2}{8 - 2} = \frac{4}{6}$$ Simplify the fraction: $$m_{AB} = \frac{\cancel{4}}{\cancel{6}} = \frac{2}{3}$$ 4. **Calculate gradient of $CD$:** Points: $C(6,8)$ and $D(0,4)$ $$m_{CD} = \frac{4 - 8}{0 - 6} = \frac{-4}{-6}$$ Simplify the fraction: $$m_{CD} = \frac{\cancel{-4}}{\cancel{-6}} = \frac{2}{3}$$ 5. **Interpretation:** Both sides $AB$ and $CD$ have the same gradient $\frac{2}{3}$, which confirms that these opposite sides of the parallelogram are parallel as expected. **Final answers:** $$\text{gradient of } AB = \frac{2}{3}$$ $$\text{gradient of } CD = \frac{2}{3}$$