Mathematics: P is the point (16,9) and Q is the point (22. 24). (a) Find the equation of the line perpendicular to PQ that passes through the point (5, 1). Give your answer in the form y=mx+c. (b) N is the point on PQ such that PN=2NQ. Find the co-ordinates of N.

2 inchesStep-by-step explanation:idk how but it is...

P is the point (16,9) and Q is the point (22. 24). - id169222412393808, JoeAnthony3293

useralreadytaken

05.05.2021

see explanation

Step-by-step explanation:

(a)

The equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

Calculate m using the slope formula

m = $\frac{y_{2}-y_{1} }{x_{2}-x_{1} }$

with (x₁, y₁ ) = (P(16, 9) and (x₂, y₂ ) = Q(22, 24)

$m_{PQ}$ = $\frac{24-9}{22-16}$ = $\frac{15}{6}$ = $\frac{5}{2}$

Given a line with slope m then the slope of a line perpendicular to it is

$m_{perpendicular}$ = - $\frac{1}{m}$ = - $\frac{1}{\frac{5}{2} }$ = - $\frac{2}{5}$ , then

y = - $\frac{2}{5}$ x + c ← is the partial equation

To find c substitute (5, 1) into the partial equation

1 = - 2 + c ⇒ c = 1 + 2 = 3

y = - $\frac{2}{5}$ x + 3 ← equation of perpendicular line

(b)

Given PN = 2NQ , then N sections PQ in the ratio 2 : 1

Using the Section formula to find coordinates of N

N = ( $\frac{16(1)+22(2)}{1+2}$ , $\frac{9(1)+24(2)}{1+2}$ )

= ( $\frac{16+44}{3}$ , $\frac{9+48}{3}$ )

= ( $\frac{60}{3}$ , $\frac{57}{3}$ )

= (20, 19 )

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