## EXERCISE 7.2

#### Page No 167:

#### Question 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 obtain

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).

#### 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.

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 runs the distance AD on the 2^{nd }line and posts a green flag. Preet runs 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?

#### Answer:

It can be observed that Niharika posted the green flag at of the distance AD i.e., m from the starting point of 2^{nd} line. Therefore, the coordinates of this point G is (2, 25).

Similarly, Preet posted red flag at of the distance AD i.e., m from the starting point of 8^{th} 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 5^{th} line.

#### Question 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.

#### 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.

#### Answer:

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

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*.

#### Answer:

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)

#### Answer:

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

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.

#### Answer:

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.

#### 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.

#### Answer:

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 = (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.