**Coordinate Geometry Exercise 7.2**

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

**Answer**

Let P(*x*, *y*) be the required point. Using the section formula, we get

Therefore, the point is (1, 3).

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

**Answer**

Let P (*x*_{1}, *y*_{1}) and Q (*x*_{2}, *y*_{2}) are the points of trisection of the line segment joining the given points i.e., AP = PQ = QB

Therefore, point P divides AB internally in the ratio 1:2.

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 the following 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 flagexactly halfway between the line segment joining the two flags, where should she post her flag?

**Answer**

It can be observed that Niharika posted the green flag at 1/4th of the distance AD i.e., (1×100/4)m = 25m from the starting point of 2nd line. Therefore, the coordinates of this point G is (2, 25).

Similarly, Preet posted red flag at 1/5 of the distance AD i.e., (1×100/5) m = 20m from the starting point of 8th line. Therefore, the coordinates of this point R are (8, 20).

Distance between these flags by using distance formula = GR

Therefore, Rashmi should post her blue flag at 22.5m on 5th line.

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

**Answer**

Let the ratio in which the line segment joining ( -3, 10) and (6, -8) is divided by point ( -1, 6) be k : 1.Therefore, -1 = 6*k*-3/*k*+1

-*k* - 1 = 6*k* -3

7*k* = 2

*k* = 2/7

Therefore, the required ratio is 2:7.

5. Find the ratio in which the line segment joining A (1, - 5) and B (- 4, 5) is divided by the x-axis. Also find the coordinates of the point of division.

**Answer**

Let the ratio in which the line segment joining A (1, - 5) and B ( - 4, 5) is divided by x-axisbe.

Therefore, the coordinates of the point of division is (-4*k*+1/*k*+1, 5*k*-5/*k*+1).

We know that y-coordinate of any point on x-axis is 0.

∴ 5*k*-5/*k*+1 = 0

Therefore, *x*-axis divides it in the ratio 1:1.

6. If (1, 2), (4, *y*), (*x*, 6) and (3, 5) are the vertices of a parallelogram taken in order, find *x *and *y*.

**Answer**

Let A,B,C and D be the points (1,2) (4,*y*), (*x*,6) and (3,5) respectively.

Mid point of diagonal AC is

and Mid point of Diagonal BD is

Since the diagonals of a parallelogram bisect each other, the mid point of AC and BD are same.

∴ *x*+1/2 = 7/2 and 4 = 5+*y*/2

⇒ *x* + 1 = 7 and 5 + *y* = 8

⇒ *x* = 6 and *y* = 3

7. Find the coordinates of a point A, where AB is the diameter of circle whose centre is (2, - 3) and B is (1, 4).

**Answer**

Let the coordinates of point A be (*x*, *y*).

Mid-point of AB is (2, - 3), which is the center of the circle.

⇒ *x* + 1 = 4 and *y + *4 = -6

⇒ *x* = 3 and *y* = -10

Therefore, the coordinates of A are (3,-10).

8. If A and B are (–2, –2) and (2, –4), respectively, find the coordinates of P such that AP = 3/7 AB and P lies on the line segment AB.

**Answer**

The coordinates of point A and B are (-2,-2) and (2,-4) respectively.

Since AP = 3/7 AB

Therefore, AP:PB = 3:4

Point P divides the line segment AB in the ratio 3:4.

9. Find the coordinates of the points which divide the line segment joining A (- 2, 2) and B (2, 8) into four equal parts.

**Answer**

From the figure, it can be observed that points X, Y, Z are dividing the line segment in a ratio 1:3, 1:1, 3:1 respectively.

10. Find the area of a rhombus if its vertices are (3, 0), (4, 5), (-1, 4) and (-2,-1) taken in order. [Hint: Area of a rhombus = 1/2(product of its diagonals)]

**Answer**

Let (3, 0), (4, 5), ( - 1, 4) and ( - 2, - 1) are the vertices A, B, C, D of a rhombus ABCD.

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