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Question 1:

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

Solution 1:

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

$x=\frac{2\times 4+3\times (-1)}{2+3}$

$=\frac{8-3}{5}=1$

$y=\frac{2\times (-3)+3\times 7}{2+3}$

$=\frac{-6+21}{5}=3$

Therefore, the point is (1, 3).

Comments

Tasneem

Question 2:

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

Solution 2:

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.

$x_{1}=\frac{1\times (-2)+2\times4}{1+2}$ $=\frac{-2+8}{3}=2$

$y_{1}=\frac{1\times (-3)+2\times (-1)}{1+2}$ $=\frac{-3-2}{3}=\frac{-5}{3}$

Therefore, $P(x_{1},y_{1})=\left ( 2,-\frac{5}{3} \right )$

Point Q divides AB internally in the ratio 2:1.

$x_{2}=\frac{2\times (-2)+1\times4}{2+1}$ $=\frac{-4+4}{3}=0$

$y_{2}=\frac{2\times (-3)+1\times (-1)}{2+1}$ $=\frac{-6-1}{3}=\frac{-7}{3}$

Therefore, $Q(x_{2},y_{2})=\left ( 0,-\frac{7}{3} \right )$

Comments

Tasneem

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 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD, as shown in the figure. Niharika runs $\frac{1}{4}th$ the distance AD on the 2nd line and posts a green flag. Preet runs $\frac{1}{5}th$ 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 3:

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

$\sqrt{(8-2)^{2}+(25-20)^{2}}= \sqrt{(36+25)}=\sqrt{61}m$

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).

$x=\frac{2+8}{2}=5, y=\frac{25+20}{2}=22.5$

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

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

Comments

Tasneem

Question 4:

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

Solution 4:

Let the ratio in which the line segment joining (−3, 10) and (6, −8) is divided by point (−1, 6) be k:1

Therefore, $-1=\frac{6k-3}{k+1}$

$-k-1=6k-3, k=\frac{2}{7}$

Therefore the ratio is 2:7

Comments

Tasneem

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.

Solution 5:

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 .

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

$\therefore \frac{5k-5}{k+1}=0$

k=1

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

Division point = $\left ( \frac{-4(1)+1}{1+1}, \frac{5(1)-5}{1+1} \right )$

$\left ( \frac{-4+1}{2}, \frac{5-5}{2} \right )=\left ( \frac{-3}{2},0 \right )$

Comments

Tasneem

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.

Solution 6:

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

$\left ( \frac{1+x}{2}, \frac{2+6}{2} \right )=\left ( \frac{x+1}{2},4 \right )$

If O is the mid-point of BD, then the coordinates of O are

$\left ( \frac{4+3}{2}, \frac{5+y}{2} \right )=\left ( \frac{7}{2},\frac{5+y}{2} \right )$

Since both the coordinates are of the same point O,

$\therefore \frac{x+1}{2}=\frac{7}{2}$ and $4= \frac{5+y}{2}$

$x+1=7, 5+y=8$

$x=6, y=3$

Comments

Tasneem

Question 7:

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

Solution 7:

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

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

$\therefore \left (2,-3\right )=\left ( \frac{x+1}{2}, \frac{y+4}{2} \right )=-3$

$x+1=4, y+4=-6$

$x=3, y=-10$

Hence the coordinates of A are $(3,-10)$

Comments

Tasneem

Question 8:

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

Solution 8:

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.

Coordinates of P = [3x2+4x(-2)]/(3+4)], [3x(-4)+4x2]/(3+4)]

= [6-8]/7, [-12-8]/7

= [-2/7] , [-20,7]

Comments

Reetika

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.

Solution 9:

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.

Coordinates of X = [1x2+3x(-2)]/(1+3) , [1x8+3x2]/(1+3)

= [-1,7]

Coordinates of Y = [2+(-2)]/2 , [2+8]/2

= [0,5]

Coordinates of Z = [3x2+1x(-2)]/(3+1) , [3x8+1x2]/(3+1)

= [1,13/2]

Comments

Reetika

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 = $\frac{1}{2}$ (product of its diagonals)]

Solution 10:

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

Length of diagonal AC = 4 $\sqrt{2}$

Length of diagonal BD = 6 $\sqrt{2}$

Therefor, area of rhombus ABCD = 1/2 x 4 $\sqrt{2}$ x 6 $\sqrt{2}$

= 24 square units.

Comments

Reetika

All Exercises - Chapter 7 - Coordinate Geometry

Exercise 7.1

Exercise 7.2

Exercise 7.3

Exercise 7.4

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Learn Exercise 7.2 with Free Lessons & Tips

All Exercises - Chapter 7 - Coordinate Geometry

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