Question

The endpoints of ab are a(1,4) and b(6,-1). If point c divides ab in the ratio 2:3, the coordinates of c are ?. If point d divides ac in the ratio 3:2, the coordinates of d are ?.

Answer 1

Answer:  The co-ordinates of point C are (3, 2) and the co-ordinates of the point D are $\left(\dfrac{11}{5},\dfrac{14}{5}\right).$

Step-by-step explanation:  Given that the endpoints of a line segment AB are  A(1,4) and B(6,-1).

We are to find the co-ordinates of a point C that divides the line segment AB in the ratio 2 : 3.

Also, to find the co-ordinates of the point D that divides the line segment AC in the ratio 3 : 2.

We know that

the co-ordinate of a point that divides a line segment with endpoints (p. q) and (r, s) in the ratio m : n is given by

$\left(\dfrac{mr+np}{m+n},\dfrac{ms+nq}{m+n}\right).$

Therefore, the co-ordinates of point C will be

$C\\\\\\=\left(\dfrac{2\times 6+3\times 1}{2+3},\dfrac{2\times (-1)+3\times 4}{2+3}\right)~~~~~~~~~~[\textup{here, m : n = 2 : 3}]\\\\\\=\left(\dfrac{12+3}{5},\dfrac{-2+12}{5}\right)\\\\\\=\left(\dfrac{15}{5},\dfrac{10}{5}\right)\\\\=(3,2).$

And, the co-ordinates of the point D will be

$D\\\\\\=\left(\dfrac{3\times 3+2\times 1}{3+2},\dfrac{3\times 2+2\times 4}{3+2}\right)~~~~~~~~~~[\textup{here, m : n = 3 : 2}]\\\\\\=\left(\dfrac{9+2}{5},\dfrac{6+8}{5}\right)\\\\\\=\left(\dfrac{11}{5},\dfrac{14}{5}\right).$

Thus, the co-ordinates of point C are (3, 2) and the co-ordinates of the point D are $\left(\dfrac{11}{5},\dfrac{14}{5}\right).$

Answer 2

Answer:

see explanation

Step-by-step explanation:

Find the coordinates of c using the section formula

$x_{c}$ = $\frac{(3(1)+(2(6)}{2+3}$ = $\frac{3+12}{5}$ = 3

$y_{c}$ = $\frac{(3(4))+(2(-1))}{2+3}$ = $\frac{12-2}{5}$ = 2

coordinates of c = (3, 2)

Similarly to find the coordinates of d

$x_{d}$ = $\frac{x(2(1))+(3(3))}{3+2}$ = $\frac{2+9}{5}$ = $\frac{11}{5}$

$y_{d}$ = $\frac{(2(4))+(3(2))}{3+2}$ = $\frac{8+6}{5}$ = $\frac{14}{5}$

coordinates of d = ( $\frac{11}{5}$, $\frac{14}{5}$)