7. COORDINATES GEOMETRY (NCERT)
NCERT
7. COORDINATES GEOMETRY
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Chapter 7 . Coordinate Geometry |
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Exercise 7.1 complete solution Exercise 7.2 complete solution Exercise 7.3 complete solution Exercise 7.4 (Optional) complete solution |
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Distance formula : 1. The distance between |
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Section formula : i.e. , i.e., 5. If |
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Area of a Triangle : 6. The area of the triangle formed by the points 7. The points |
and 
is given by
from the origin 
is given by 
which divides the line segment joining the points 
and 
internally in the ratio 
are given by 
and 
and 
are the vertices of 
, then the coordinates of the centroid is 
EXERCISE 7.1
1. Find the distance between the following pairs of points :
(i) 
(ii) 
(iii) 
Solution : (i)
Let 
and 
are two points .
Using distance formula , we have
units
(ii)
Let 
and 
are two points .
Using distance formula , we have
units
(iii)
Let 
and 
are two points .
Using distance formula , we have
units
2. Find the distance between the points 
and 
. Can you now find the distance between the two towns A and B discussed in Section 7.2 .
Solution : Let 
and 
are two points .
Using distance formula , we have
units
ghghghghghghgh
3. Determine if the points 
and 
are collinear .
Solution : Let 
and 
are three points respectively .
Using distance formula , we have
units
units
units
So , 
.
Therefore, the points and are not collinear .
4. Check whether 5
and 
are the vertices of an isosceles triangle .
Solution : Let 
and 
are the vertices of any triangle respectively .
Using distance formula , we have
units
units
units
So, 
Therefore ,the points and are the vertices of an isosceles triangle .
5. In a classroom, 4 friends are seated at the points A , B , C and D as shown in Fig. 7.8 . Champa and Chemeli walk into the class and after observing for a few minutes Champa asks Chameli ‘‘ Don’t you think ABCD is a square ? Chameli disagrees . Using distance formula , find which of them is correct .
Solution:
6. Name the type of quadrilateral formed, if any , by the following points, and give reasons for your answers :
(i) 
(ii) 
(iii) 
Solution: (i)
Let 
and 
are the vertices of the quadrilateral respectively.
Using distance formula , we have
units
units
units
units
units
units
So, 
and 
Therefore , 
and 
are vertices of the square .
(ii)
Solution: Let 
and 
are the vertices of the quadrilateral respectively.
Using distance formula , we have
units
units
units
units
So,
Therefore , and are not of the vertices of quadrilateral .
(iii)
Solution: Let 
and 
are the vertices of the quadrilateral respectively.
Using distance formula , we have
units
units
units
units
units
units
So, 
, 
and
Therefore , and are vertices of the parallelogram .
7. Find the point on the 
-axis which is equidistant from 
and 
.
Solution : Let, 
is equidistant from the points A(2 , – 5 ) and B (– 2 ,9) .
Given , -axis , i.e.,
.
A/Q , 
Therefore , the coordinate of the point P is 
.
8. Find the values of 
for which the distance between the points 
and
is 10 units .
Solution : We have , 
or 
Thus, the value of are – 9 and 3 .
9. If 
is equidistant from 
and 
, find the values of . Also find the distance QR and PR .
Solution : Since is equidistant from and 
.
A/Q, 
The distance of and 
is
units
The distance of and 
is
units
The distance of and is
units
The distance of and is
units
10. Find a relation between 
and 
such that the point
is equidistant from the point 
and 
.
Solution : Given ,the point 
is equidistant from
the point 
and 
.
We have , 
[Squaring both side]
EXERCISE 7.2
1. Find the coordinates of the point which divides the join of 
and 
in the ratio 
.
2. Find the coordinates of the points of trisection of the line segment joining 
and 
.
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 flowers pots have been placed at a distance of 1 m from each other along AD , as shown in Fig. 7.12 . Niharika runs 
th the distance AD , on the 
line and posts a green flag . Preet runs 
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 jointing the two flags, where should she post her flag ?
4. Find the ratio in which the line segment joining the points 
and 
is divided by
.
5. Find the ratio in which the line segment joining 
and 
is divided by the -axis . Also , find the coordinates of the point of division .
6. If 
and 
are the vertices of a parallelogram taken in order, find and 
.
7. Find the coordinates of a point A , where AB is the diameter of a circle whose centre is 
and 
is 
.
8. If A and B are 
and 
, respectively, find the coordinates of P such that 
and P lies on the line segment AB .
9. Find the coordinates of the points which divide the line segment joining 
and 
into four equal parts .
10. Find the area of a rhombus if its vertices are 
and 
taken in order . [Hint : Area of a rhombus 
(product of its diagonals)]
EXERCISE 7.3
1. Find the area of the triangle whose vertices are :
(i) 
(ii) 
2. In each of the following find the value of ‘ ’ , for which the points are collinear :
(i) 
(ii) 
3. Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are 
and
.Find the ratio of this area to the area of the given triangle .
4. Find the area of the quadrilateral whose vertices, taken in order are 
and
.
5. You have studied in Class IX , (Chapter 9 , Example 3) , that a median of a triangle divides it into two triangles of equal areas . Verify this result for 
whose vertices are 
and 
.
EXERCISE 7.4 (Optional)*
1. 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).
2. Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear.
3. Find the centre of a circle passing through the points (6, – 6), (3, – 7) and (3, 3).
4. The two opposite vertices of a square are (–1, 2) and (3, 2). Find the coordinates of the other two vertices.
5. The Class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Sapling of Gulmohar are planted on the boundary at a distance of 1m from each other. There is a triangular grassy lawn in the plot as shown in the Fig. 7.14. The students are to sow seeds of flowering plants on the remaining area of the plot.
(i) Taking A as origin, find the coordinates of the vertices of the triangle.
(ii) What will be the coordinates of the vertices of ∆ PQR if C is the origin? Also calculate the areas of the triangles in these cases. What do you observe?
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, such that 
⋅ Calculate the area of the ∆ ADE and compare it with the area of ∆ ABC. (Recall Theorem 6.2 and Theorem 6.6).
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
(iii) Find the coordinates of points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1 .
(iv) What do yo observe?
[Note : The point which is common to all the three medians is called the centroid and this point divides each median in the ratio 2 : 1.]
(v) If A( 
) , B( 
) and C( 
) are the vertices of ∆ ABC, find the coordinates of the centroid of the triangle.
8. ABCD is a rectangle formed by the points A(–1, –1), B(– 1, 4), C(5, 4) and D(5, – 1). P, Q , R 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.
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