- Find the coordinates of the point which divides the join of (− 1, 7) and (4, − 3) in the ratio 2:3.
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).
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?
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).
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
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.
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)
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.
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.
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)]
Let (3, 0), (4, 5), (−1, 4) and (−2, −1) are the vertices A, B, C, D of a rhombus ABCD.