Question

𝑃 and 𝑄 are the points (3,1) and (−2,−9) respectively. Calculate:

a. The distance 𝑃𝑄

b. The midpoint of the line joining 𝑃 to 𝑄

c. The gradient of the line 𝑃𝑄

d. The gradient of a line perpendicular to 𝑃𝑄

Answer 1

a.

$$PQ=\sqrt{(-2-3)^2+(-9-1)^2}=\sqrt{125}$$

$$=5\sqrt{5}\ (units)$$

b.

$$x_M=\dfrac{x_P+x_Q}{2}=\dfrac{3-2}{2}=\dfrac{1}{2}$$

$$y_M=\dfrac{y_P+y_Q}{2}=\dfrac{1-9}{2}=-4$$

$M(1/2, -4)$

c.

$$gradient_1=m_1=\dfrac{y_Q-y_P}{x_Q-x_P}=\dfrac{-9-1}{-2-3}=2$$

$gradient_1=2$

d.

$$gradient_2=-\dfrac{1}{gradient_1}=-\dfrac{1}{2}$$

$gradient_2=-\dfrac{1}{2}$