Question 1: Find the coordinates of the point which divides the join of (-1, 7) and (4, -3) in the ratio 2 : 3.

**Solution:** A = (-1, 7) and B = (4, -3)

So, `x_1 = -1, y_1 = 7, x_2 = 4` and `y_2 = -3`

We also have; `m_1 = 2` and `m_2 = 3`

Let us assume P is the point of division

Coordinates of P can be calculated as follows by using section formula:

`x=(m_1x_2+m_2x_1)/(m_1+m_2)`

`=(2xx4+3xx-1)/(5)`

`=(8-3)/(5)=5/5=1`

`y=(m_1y_2+m_2y_1)/(m_1+m_2)`

`=(2xx-3+3xx7)/(5)`

`=(-6+21)/(5)=15/5=3`

Hence, P = (1, 3)

Question 2: Find the coordinates of the points of trisection of the line segment joining (4, -1) and (-2, -3).

**Solution:** We have A = (4, -1) and B = (-2, -3)

Here; `x_1 = 4`, `y_1 = -1`, `x_2 = -2` and `y_2 = -3`

Let us assume that points C and D divide line segment AB into three equal parts so that AC = CD = DB

In that case, point C divides AB in ratio 1 : 2 and point D divides AB in ratio 2 : 1

For point C; `m_1 = 1` and `m_2 = 2`

For point D; `m_1 = 2` and `m_2 = 1`

Coordinates of point C can be calculated as follows:

`x=(m_1x_2+m_2x_1)/(m_1+m_2)`

`=(1xx-2+2xx4)/(3)`

`=(-2+8)/(3)=6/3=2`

`y=(m_1y_2+m_2y_1)/(m_1+m_2)`

`=(1xx-3+2xx-1)/(3)`

`=(-3-2)/(3)=-5/3`

Coordinates of point D can be calculated as follows:

`x=(m_1x_2+m_2x_1)/(m_1+m_2)`

`=(2xx-2+1xx4)/(3)`

`=(-4+4)/(3)=0`

`y=(m_1y_2+m_2y_1)/(m_1+m_2)`

`(2xx-3+1xx-1)/(3)`

`=(-6-1)/(3)=-7/3`

Hence; `C = (2, -5/3)` and `D = (0, -7/3)`

Question 3: To conduct sports day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots have been placed at a distance of 1 m from each other along AD, as shown in figure. Niharika runs 1/4th the distance AD on the 2nd line and posts a green flag. Preet runs 1/5th the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?

**Solution:** Coordinates for green flag A = (2, 25)

Coordinates for red flag B = (8, 20)

Using distance formula, length of AB can be calculated as follows:

`AB=sqrt((x_2-x_2)^2+(y_2-y_1)^2)`

`=sqrt((8-2)^2+(20-25)^2)`

`=sqrt(6^2+(-5)^2))`

`=sqrt(36+25)=sqrt61\m`

Coordinates for blue flag can be calculated by using midpoint formula:

`x=(x_1+x_2)/(2)`

`=(8+2)/(2)=5`

`y=(y_1+y_2)/(2)`

`=(25+20)/(2)=45/2`

Hence; `C = (5, 45/2)`

Question 4: Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by (-1, 6).

**Solution:** Answer: `x_1 = -3`, `y_1 = 10`, `x_2 = 6`, `y_2 = -8`, `x_3 = -1` and `y_3 = 6`

Let us assume that point (-1, 6) divides the line segment in ratio m1 : m2

Then x coordinate for this point can be given as follows:

`-1=(-3m_2+6m_1)/(m_1+m_2)`

Or, `-(m_1+m_2)=-3m_2+6m_1`

Or, `m_1+m_2=3m_2-6m_1`

Or, `7m_1=2m_2`

Or, `m_1/m_2=2/7=2:7`

The y coordinate for this point can be given as follows:

`6=(10m_2-8m_1)/(m_1+m_2)`

Or, `6m_1+6m_2=10m_2-8m_1`

Or, `14m_1=4m_2`

Or, `m_1/m_2=4/14=2:7`

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