Question 1:
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).
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.
Point Q divides AB internally in the ratio 2:1.
Therefore, the point is (1, 3).
Question 2:
Find the coordinates of the points of trisection of the line segment joining (4, − 1) and (− 2, − 3).
Therefore, point P divides AB internally in the ratio 1:2.
Point Q divides AB internally in the ratio 2:1.
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 the following figure. Niharika runs1/4th the distance AD on the 2nd line and posts a green flag. Preet runs1/5th 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?
It can be observed that Niharika posted the green flag at 1/4 of the distance AD i.e.,
m 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.,
m 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
=
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).
Therefore, Rashmi should post her blue flag at 22.5m on 5th line.
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
If O is the mid-point of BD, then the coordinates of O are
Since both the coordinates are of the same point O,
m 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.,
m 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
=
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).
Therefore, Rashmi should post her blue flag at 22.5m on 5th line.
Question 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.
Question 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-axisbe.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 =
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 =
Question 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
If O is the mid-point of BD, then the coordinates of O are
Since both the coordinates are of the same point O,
Question 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.
Mid-point of AB is (2, −3), which is the center of the circle.
Question 8:
If A and B are (− 2, − 2) and (2, − 4), respectively, find the coordinates of P such that
and P lies on the line segment AB.
The coordinates of point A and B are (−2, −2) and (2, −4) respectively.
Since ,
Therefore, AP: PB = 3:4
Point P divides the line segment AB in the ratio 3:4.
From the figure, it can be observed that points P, Q, R are dividing the line segment in a ratio 1:3, 1:1, 3:1 respectively.
Let (3, 0), (4, 5), (−1, 4) and (−2, −1) are the vertices A, B, C, D of a rhombus ABCD.
and P lies on the line segment AB.
The coordinates of point A and B are (−2, −2) and (2, −4) respectively.
Since ,
Point P divides the line segment AB in the ratio 3:4.
Question 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 observed that points P, Q, R are dividing the line segment in a ratio 1:3, 1:1, 3:1 respectively.
Question 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)]
Let (3, 0), (4, 5), (−1, 4) and (−2, −1) are the vertices A, B, C, D of a rhombus ABCD.
