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)

## Step1: <br />First, we need to find the vectors \( \overrightarrow{OM} \) and \( \overrightarrow{ON} \). Since the points \( M \) and \( N \) divide \( OB \) and \( CA \) 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 \( \overrightarrow{OM} = \frac{1}{1+4} \overrightarrow{OB} \) and \( \overrightarrow{ON} = \frac{4}{1+4} \overrightarrow{OA} \).<br /><br />## Step2:<br />Substitute \( \overrightarrow{OB} = \mathbf{b} \) and \( \overrightarrow{OA} = \mathbf{a} \) into the formulas from Step1 to get \( \overrightarrow{OM} = \frac{1}{5} \mathbf{b} \) and \( \overrightarrow{ON} = \frac{4}{5} \mathbf{a} \).<br /><br />## Step3:<br />To find \( \overrightarrow{MN} \), we subtract \( \overrightarrow{OM} \) from \( \overrightarrow{ON} \). This gives \( \overrightarrow{MN} = \overrightarrow{ON} - \overrightarrow{OM} = \frac{4}{5} \mathbf{a} - \frac{1}{5} \mathbf{b} \).<br /><br />### \( \overrightarrow{MN} = \frac{4}{5} \mathbf{a} - \frac{1}{5} \mathbf{b} \)<br /><br />## Step4:<br />Next, we find the vectors \( \overrightarrow{OS} \) and \( \overrightarrow{OT} \) using the same method as in Step1. We get \( \overrightarrow{OS} = \frac{1}{1+4} \overrightarrow{OC} \) and \( \overrightarrow{OT} = \frac{4}{1+4} \overrightarrow{OA} \).<br /><br />## Step5:<br />Since \( OC = OB = \mathbf{b} \) and \( OA = \mathbf{a} \), we substitute these into the formulas from Step4 to get \( \overrightarrow{OS} = \frac{1}{5} \mathbf{b} \) and \( \overrightarrow{OT} = \frac{4}{5} \mathbf{a} \).<br /><br />## Step6:<br />To find \( \overrightarrow{ST} \), we subtract \( \overrightarrow{OS} \) from \( \overrightarrow{OT} \). This gives \( \overrightarrow{ST} = \overrightarrow{OT} - \overrightarrow{OS} = \frac{4}{5} \mathbf{a} - \frac{1}{5} \mathbf{b} \).<br /><br />### \( \overrightarrow{ST} = \frac{4}{5} \mathbf{a} - \frac{1}{5} \mathbf{b} \)<br /><br />## Step7:<br />Finally, we can see that \( \overrightarrow{MN} = \overrightarrow{ST} \), which means that the lines \( MN \) and \( ST \) bisect each other.