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 point C are (
,
).

If point D divides AC in the ratio 3 : 2, the coordinates of point D are (
,
).

Answer 1

C is
`(3,2)`

D is
`(2.2,2.8)`

Explanation:

(i) We need to find the coordinate of point C, which divides A(1,4) and B(6,-1) in the ratio 2:3

We know that the coordinate of a point
`(x,y)` dividing a line segment joining
`(x_(1),y_(1)) \:\text{and} (x_(2),y_(2))` in the ratio m:n is given by

`x=(mx_(2)+nx_(1))/(m+n)`,
`y=(my_(2)+ny_(1))/(m+n)`

Let the coordinate of point C be
`(x,y)`

Here, the ratio is 2:3

So,
`x=(2(6)+3(1))/(2+3)`

`x=(15)/(5)`

`x=3`

`y=(2(-1)+3(4))/(2+3)`

`y=(-2+12)/(5)`

`y=(10)/(5)`

`y=2`

Hence, coordinate of C is
`(3,2)`

(ii) We need to find the coordinate of point D, which divides A(1,4) and C(3,2) in the ratio 3:2

We know that the coordinate of a point
`(x,y)` dividing a line segment joining
`(x_(1),y_(1)) \:\text{and} (x_(2),y_(2))` in the ratio m:n is given by

`x=(mx_(2)+nx_(1))/(m+n)`,
`y=(my_(2)+ny_(1))/(m+n)`

Let the coordinate of point D be
`(x_(3),y_(3))`

Here, the ratio is 2:3

So,
`x_(3)=(3(3)+2(1))/(3+2)`

`x_(3)=(11)/(5)`

`x_(3)=2.2`

`y_{3=(3(2)+2(4))/(3+2)`

`y_{3=(6+8)/(5)`

`y_{3=(14)/(5)`

`y_{3=2.8`

Hence, coordinate of D is
`(2.2,2.8)`

Answer 2

C(3, 2)

D(2.2, 2.8)

Explanation:

If point
`M(x,y)` divides the segment AB with endpoints
`A(x_1,y_1)` and
`B(x_2,y_2)` in the ratio
`m:n,` then its coordinates are

`x=(n\cdot x_1+m\cdot x_2)/(m+n)\\ \\y=(n\cdot y_1+m\cdot y_2)/(m+n)`

You are given A(1,4) and B(6,-1).

1. If point C divides AB in the ratio 2 : 3, the coordinates of point C are

`x=(3\cdot 1+2\cdot 6)/(2+3)=(3+12)/(5)=(15)/(5)=3\\ \\y=(3\cdot 4+2\cdot (-1))/(2+3)=(12-2)/(5)=(10)/(5)=2`

C(3, 2)

2. If point D divides AC in the ratio 3 : 2, the coordinates of point D are

`x=(2\cdot 1+3\cdot 3)/(3+2)=(2+9)/(5)=(11)/(5)=2.2\\ \\y=(2\cdot 4+3\cdot 2)/(3+2)=(8+6)/(5)=(14)/(5)=2.8`

D(2.2, 2.8)