13. OACB is a parallelogram. overrightarrow (OA)=a and overrightarrow (OB)=b The points M, S N and T divide OB, BC, CA and AO in the ratio 1:4 respectively. The lines ST and MN intersect at the point D. (a) Express overrightarrow (MN) n terms of a and b. (b) Express overrightarrow (ST) in terms of a and b. (c) Show that the lines MN and ST bisect one another. (2) (2) (9) (Total 13 marks)

Question

Answer 1

(a)

(b)

(c) The lines

and

bisect each other.

Answer 2

## Step1: First, we need to find the vectors

and

. Since the points

and

divide

and

in the ratio 1:4 respectively, we can use the formula for dividing a line segment in a given ratio to find these vectors. The formula is

and

.## Step2:Substitute

and

into the formulas from Step1 to get

and

.## Step3:To find

, we subtract

from

. This gives

.###

## Step4:Next, we find the vectors

and

using the same method as in Step1. We get

and

.## Step5:Since

and

, we substitute these into the formulas from Step4 to get

and

.## Step6:To find

, we subtract

from

. This gives

.###

## Step7:Finally, we can see that

, which means that the lines

and

bisect each other.

Question