Question

If $$\mathrm{A}(2, -3), \mathrm{B}(-5, 1), \mathrm{C}(7, -1)$$ and $$\mathrm{D}(0, k)$$ be four points such that AB is parallel to CD, find the value of $$k$$.

Answer 1

Slope of a line passing through two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ is given by $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Slope of $$\mathrm{AB}$$:

$$m_{\mathrm{AB}} = \frac{1 - (-3)}{-5 - 2}$$

$$= \frac{1 + 3}{-7}$$

$$= -\frac{4}{7}$$

Slope of $$\mathrm{CD}$$:

$$m_{\mathrm{CD}} = \frac{k - (-1)}{0 - 7}$$

$$= \frac{k + 1}{-7}$$

Since $$\mathrm{AB} \parallel \mathrm{CD}$$, their slopes are equal:

$$m_{\mathrm{AB}} = m_{\mathrm{CD}}$$

$$-\frac{4}{7} = \frac{k + 1}{-7}$$

$$k + 1 = 4$$

$$k = 3$$