Question

Find the coordinates of three points that divide the line segment joining P(-4,7) and Q(10,-9) into four parts of equal

Answer 1

Answer:

A = (-1/2 , 3).

B = (3 , - 1).

C = (13/2 , - 5)

Step-by-step explanation:

Given,

P = (-4 , 7)

Q = (10 , - 9)

Three co-ordinates divide the line PQ into 4 equal parts

Let, the 3 co-ordinates be A , B , C

To Find :-

Value of the three co-ordinates (A , B , C)

How To Do :-

As the 3 co-ordinates divide the line into four equal parts , we need to find the ratio that A divides PQ and B divides PQ and C divides PQ. and we need to apply section(Internal division) formula to get the Co-ordinates of A , B , C.

Formula Required :-

Section(Internal division) formula :-

$=\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)$

Solution :-

As all the co-ordinates divided the PQ into four equal parts :-

A divides PQ in the ratio :- 1 : 3

B divides PQ in the ratio :- 2 : 2 = 1 : 1

C divides PQ in the ratio :- 3 : 1

Finding the co-ordinates of A :-

Let,

m : n = 1 : 3

P = (-4 , 7)

x_1 = -4 , y_1 = 7

Q = (10 , -9)

x_2 = 10 , y_2 = - 9

$A = \left(\dfrac{1(10)+3(-4)}{1+3},\dfrac{1(-9)+3(7)}{1+3}\right)\\\\=\left(\dfrac{10-12}{4},\dfrac{-9+21}{4}\right)$

= (-2/4 , 12/4)

= (-1/2 , 3)

Co-ordinates of A = (-1/2 , 3).

Finding Co-ordinates for B :-

m : n = 1 : 1

P = (-4 , 7)

Let,

x_1 = -4 , y_1 = 7

Q = (10 , -9)

Let,

x_2 = 10 , y_2 = -9

$B=\left(\dfrac{1(10)+1(-4)}{1+1},\dfrac{1(-9)+1(7)}{1+1}\right)$

$=\left(\dfrac{10-4}{2},\dfrac{-9+7}{2}\right)$

$=\left(\dfrac{6}{2},\dfrac{-2}{2}\right)$

= (3 , - 1)

Co-ordinates of B = (3 , - 1).

Finding co-ordinates of C :-

m : n = 3 : 1

P = (-4 , 7)

Let,

x_1 = -4 , y_1 = 7

Q = (10 , -9)

Let,

x_2 = 10 , y_2 = -9

$C=\left(\dfrac{3(10)+1(-4)}{3+1},\dfrac{3(-9)+1(7)}{3+1}\right)$

$=\left(\dfrac{30-4}{4},\dfrac{-27+7}{4}\right)$

= (26/4 , -20/4)

= (13/2 , - 5)

Co-ordinates of C = (13/2 , - 5).

Note : Refer to above picture .