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

**Solution: **

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

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). **

Solution:

Let P (x1, y1) and Q (x2, y2) 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 4 𝑡ℎ the distance AD on the 2nd line and posts a green flag. Preet runs 1 5 𝑡ℎ 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:

It can be observed that Niharika posted the green flag at 1/4 of the distance AD i.e.,

*[(¼) x 100)]m = 25*

metre 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.,

*[(1/5) x 100)]m = 25*

metre 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

= √[(8 – 2)² + (25 – 20)²] = √[36+25] = √61 m

The point at which Rashmi should post her blue flag is the mid-point of the line joining these points. Let this point be A (x, y).

x = (2+8)/2 = 10/2 = 5

y = (25+20)/2 = 45/2 = 22.5

Hence A(x,y) = (5, 22.5)

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). **

Solution:

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 = (6k-3)/(k+1)

-k-1 = 6k – 3

7k = 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**

Solution:

Let the ratio in which the line segment joining A (1, −5) and B (−4, 5) is divided by x – axis be k:1.

Therefore, the coordinates of the point of division is

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

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

Division point =

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

Solution:

Let (1, 2), (4, y), (x, 6), and (3, 5) are the coordinates of A, B, C, D vertices of a parallelogram ABCD. Intersection point O of diagonal AC and BD also divides these diagonals. Therefore, O is the mid-point of AC and BD. If O is the mid-point of AC, then the coordinates of O are

**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) **

Solution:

Let the coordinates of point A be (x, y). Mid-point of AB is (2, −3), which is the center of the circle.

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

Solution:

The coordinates of point A and B are (-2, -2) and (2, -4) respectively. Since AP = **𝐴𝑃 = 3/7𝐴𝐵**

Therefore AP:PB = 3:7

Point P divides the line AB in the ratio 3:7

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

Solution:

From the figure it can be observd that the points P, Q, R divides the line segments in the 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 = ½ (product of its diagonals)]**

Solution:

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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