Question

1. which equation represents a line that is perpendicular

to the line represented by ( y = \frac { 3 } { 2 } x + 1 )?
a) ( 3 x + 2 y = 12 )
b) ( 3 x - 2 y = 12 )
c) ( y = \frac { 3 } { 2 } x + 2 )
d) ( y = - \frac { 2 } { 3 } x + 4 )

1. directed line segment ( d e ) has endpoints ( d ( - 4, - 2 ) )

and ( e ( 1,8 ) ). point ( f ) divides ( overline { d e } ) such that ( d f : f e ) is
( 2 : 3 ). what are the coordinates of ( f )?
a) ( ( - 3,0 ) )
b) ( ( - 2,2 ) )
c) ( ( - 1,4 ) )
d) ( ( 2,4 ) )

1. point ( p ) divides the directed line segment from point ( a )

( ( - 4, - 1 ) ) to point ( b ( 6,4 ) ) in the ratio ( 2 : 3 ). the
coordinates of point ( p ) are
a) ( ( - 1,1 ) )
b) ( ( 0,1 ) )
c) ( ( 1,0 ) )
d) ( ( 2,2 ) )

Explanation:

Step1: Use the section formula

The section formula for a point \(F(x,y)\) that divides the line segment joining \(D(x_1,y_1)\) and \(E(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1=-4,y_1=-2,x_2 = 1,y_2=8,m = 2,n=3\).

Step2: Calculate the \(x\) - coordinate

Substitute the values into the \(x\) - coordinate formula:
\(x=\frac{2\times1+3\times(-4)}{2 + 3}=\frac{2-12}{5}=\frac{-10}{5}=-2\)

Step3: Calculate the \(y\) - coordinate

Substitute the values into the \(y\) - coordinate formula:
\(y=\frac{2\times8+3\times(-2)}{2 + 3}=\frac{16 - 6}{5}=\frac{10}{5}=2\)

Answer:

B. \((-2,2)\)