Question

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

Answer 1

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

$$m_1 = \frac{5 - (-5)}{-2 - 2}$$

$$= \frac{10}{-4}$$

$$= -\frac{5}{2}$$

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

$$m_2 = \frac{1 - 3}{1 - k}$$

$$= \frac{-2}{1 - k}$$

Since $$\mathrm{AB} \perp \mathrm{CD}$$, $$m_1 \times m_2 = -1$$: 

$$\left(-\frac{5}{2}\right) \times \left(\frac{-2}{1 - k}\right) = -1$$ 

$$\frac{5}{1 - k} = -1$$

$$5 = -(1 - k)$$

$$5 = -1 + k$$

$$k = 6$$