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

  exercise 7.2 cordinate geometry chapter 7 cbse ncert 10th mathematics 

section formula , midpoint formula 

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

A(1, 2),B (4, y),C (x, 6) and D(3, 5)

 

since the vertices are said to be taken in order

we take the diagonals as AC and BD

 

A(1, 2),C (x, 6)

using midpoint formula,

midpoint of AC =[{1+x}/2  , {2+6}/2 ] = [{1+x}/2  ,4 ]  

 

B (4, y), D(3, 5)

using midpoint formula,

midpoint of BD =[{4+3}/2  , {y+5}/2 ] = [7/2  ,{y+5}/2 ]  

We know that the diagonals of a parallelogram bisect each other

so the midpoints of AC and BD are the same

[{1+x}/2  ,4 ] =[7/2  ,{y+5}/2 ]  

using separate co ordinates

{1+x}/2 = 7/2

1+x=7

x=6

4={y+5}/2

8=y+5

y=3

 

 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)

 

let A=(h.k)

 B = (1,4)

 using formula for midpoint of AB,

midpoint of AB is  [ {h+1}/2 , {k+4}/2  ]

This is the same as the midpoint (2, -3)

 

[ {h+1}/2 , {k+4}/2  ] = (2, -3)

 

separate the coordinates

 

{h+1}/2 = 2

h+1 = 4

h=3

 

(k+4)/2 =   -3

k+4 = (-6)

k=-6-4

k= (-10)

so that A is (3, -10)

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ncert cbse 10th mathematics

 

co ordinate geometry chapter 7

exercise 7.4 optional exercise  

 

Determine the ratio in which the line 2x + y – 4 = 0 divides the line segment joining the points A(2, – 2) and B(3, 7).

 solution 

 

2. Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear.

solution 

 

3. Find the centre of a circle passing through the points (6, – 6), (3, – 7) and (3, 3).

solution

4. The two opposite vertices of a square are (–1, 2) and (3, 2). Find the coordinates of the other two vertices.

solution   

6. The vertices of a ∆ ABC are A(4, 6), B(1, 5) and C(7, 2). A line is drawn to intersect sides AB and AC at D and E respectively so that [AD/AB] =[AE/AC] =[1/4] Calculate the area of  ∆ ADE and compare it with the area of ∆ ABC

solution

7. Let A (4, 2), B(6, 5) and C(1, 4) be the vertices of ∆ ABC.
(i) The median from A meets BC at D. Find the coordinates of the point D.

(ii) Find the coordinates of the point P on AD such that AP : PD = 2 : 1  

solution 

 

8. ABCD is a rectangle formed by the points A(–1, –1), B(– 1, 4), C(5, 4) and

 D(5, – 1). P, Q, and S are the mid-points of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.

solution

 

exercise 7.3

 Find the area of the triangle whose vertices are

  (2, 3), (–1, 0), (2, – 4)

 solution

(ii) (–5, –1), (3, –5), (5, 2)

solution

 

2. In each of the following find the value of ‘k’, for which the points are collinear.
 

 (7, –2), (5, 1), (3, k)

 solution

(ii) (8, 1), (k, – 4), (2, –5)

solution 

 

4. Find the area of the quadrilateral whose vertices, taken in order, are (– 4, – 2), (– 3, – 5), (3, – 2) and (2, 3).

solution 

 

exercise 7.2

 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

 

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

 

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 

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

 solution

2. Find the coordinates of the points of trisection of the line segment joining

 (4, –1) and (-2,-3)

 

solution

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

solution

 

 

 

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