Chapter 7 . Coordinate Geometry |
Exercise 7.1 complete solution Exercise 7.2 complete solution Exercise 7.3 complete solution Exercise 7.4 (Optional*) complete solution |
Distance formula : 1. The distance between and is given by |
Section formula : i.e. , and i.e., and 5. If , and are the vertices of , then the coordinates of the centroid is
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Area of a Triangle : 6. The area of the triangle formed by the points , and is given by 7. The points , and are collinear ,then
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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 figure 7.2 .
Solution : Let and are two points .
Using distance formula , we have
units
Second part :
Here, the point A and B lie on the x-axis .
Iin figure , OA = 4 units and OB = 6 units
AB = OB - OA = 6 - 4 = 2 units
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: Given, the coordinates of the four friends are A(3,4) , B(6,7) , C(9,4) and D(6,1) respectively .
In figure :
Using distance formula, we have
units
units
units
Again,
units
And
units
So, AB = BC = CD = AD and AC = BD
Therefore, ABCD is a square . Chapma is correct .
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 : In given figure :
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]
1. Find the coordinates of the point which divides the join of and in the ratio .
Solution: Here, , ,
Let the coordinate of the point is P .
Using section formula , we have
And
Therefore, the coordinates of the point is (1 , 3) .
2. Find the coordinates of the points of trisection of the line segment joining and .
Solution: let the coordinates of the points are and Q .
For point P : Here, , ,
Using section formula , we have
And
For point Q : Here, , ,
And
Therefore, the coordinates of the points are 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 ?
Solution: Given, ABCD is a rectangular school ground , then the distance of the side AD = 1m × 100 = 100 m .
The distance of AD run by Niharika on the second line
Therefore, the coordinate of the point is (2 , 25) .
Again , the distance of AD run by Preet on the eighth line
Therefore, the coordinate of the point is (8 , 20) .
Using distance formula, we have
The distance between both the flags
m
Since, Rashmi has to post a blue flag exactly halfway between the line segment jointing the two flags ,i.e., Rashmi is equidistant from Niharika and Preet . let the coordinate of Rashmi is
Using section formula , we have
and
Therefore, the position of Rashmi has to post a blue flag on the 5th line at a distance of 22.5 m .
4. Find the ratio in which the line segment joining the points and is divided by .
Solution: let , the ratio be .
Here, , ,
Using section formula , we have
and
Now ,
Therefore, the ratio is 2 : 7 .
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 .
Solution: let , the ratio be and the coordinate is .
Here, ,
Using section formula , we have
and
Now,
Again,
Therefore, the ratio is 1 : 1 and the coordinate is .
6. If and are the vertices of a parallelogram taken in order, find and .
Solution: We know that , the diagonals of a parallelogram bisect each other .
A/Q, The coordinate of the mid-point of the diagonal AC = The coordinate of the mid-point of the diagonal BD .
Or
Therefore, the value of and .
7. Find the coordinates of a point A , where AB is the diameter of a circle whose centre is and is .
Solution: Here, AP = BP = Radius . So,
Let the coordinates of a point A is .
Here, ,
Using the section, we have
and
Therefore, the coordinate of the point .
8. If A and B are and , respectively, find the coordinates of P such that and P lies on the line segment AB .
Solution: let the coordinates of the point P is .
Now,
Here, ,,
Using section formula , we have
and
Therefore, the coordinate of the point P is .
9. Find the coordinates of the points which divide the line segment joining and into four equal parts .
Solution: let the coordinate of the points are and .
Here, ,
For point P :
Using section formula , We have
and
The coordinate of the point P is .
For point Q : Here,
and
The coordinate of the point Q is (0 , 5) .
For point R : Here,
and
The coordinate of the point R is .
10. Find the area of a rhombus if its vertices are and taken in order . [Hint : Area of a rhombus (product of its diagonals)]
Solution: let the vertices of the rhombus are and respectively .
Using distance formula , we have,
units
and
units
The area of a rhombus ABCD
sq. units
1. Find the area of the triangle whose vertices are :
(i) (ii)
Solution: (i)
let A(2,3) , B(– 1 , 0) and C(2 ,– 4) are the vertices of the triangle ABC respectively .
Here, ,
We know that ,
Area of
Sq. units .
Solution: (ii)
let A(– 5 ,– 1) , B(3 ,– 5) and C(5 , 2) are the vertices of the triangle ABC respectively .
Here, ,
We know that ,
Area of
Sq. units
2. In each of the following find the value of ‘k ’ , for which the points are collinear :
(i) (ii)
Solution: (i)
Here, ,
We know that,
Therefore, the value of k is 4 .
Solution: (ii)
Here, ,
We know that,
Therefore, the value of k is 3 .
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 .
Solution: let A(0 ,– 1) , B(2 , 1) and C(0 , 3) are the vertices of the triangle ABC respectively .
Here, ,
We know that ,
Area of
sq. units
Let D , E and F are the mid-point of the sides AB , BC and AC of the triangle ABC respectively .
So, the coordinate of the point D is
The coordinate of the point E is
The coordinate of the point F is
Therefore, D(1 , 0) , E(1 , 2) and C(0 , 1) are the vertices of the triangle DEF respectively .
Here, ,
Area of
sq. units
So,
4. Find the area of the quadrilateral whose vertices, taken in order are and .
Solution: let A(– 4 ,– 2) , B(– 3 , – 5) , C(3 ,– 2) and D(2,3) are the vertices of the triangle ABC respectively and join diagonal AC .Then ,we find two triangles ABC and ADC .
For triangle ABC : The vertices of the triangle ABC are A(– 4 ,– 2) , B(– 3 , – 5) and C(3 ,– 2) respectively.
Here, ,
We know that ,
Area of
Sq. units
For triangle ADC : The vertices of the triangle ABC are A(– 4 ,– 2) , D(2 , 3) and C(3 ,– 2) respectively.
Here, ,
We know that ,
Area of
Sq. units (positive value)
Therefore, Area of ABCD = Area of ABC + Area of ADC
Sq.units .
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 .
Solution: let AD is the median of a triangle ABC . The vertices of the triangle ABC are and . We find two triangles are ABD and ACD .
The coordinate of the point D is
For triangle ABD : The vertices of the triangle ABD are A( 4 ,– 6) , B(3 ,– 2) and D(4 ,0) respectively.
Here, ,
We know that ,
Area of
Sq. units (positive value only)
For triangle ACD : The vertices of the triangle ABD are A( 4 ,– 6) , B(5 , 2) and D(4 ,0) respectively.
Here, ,
We know that ,
Area of
Sq. units
Therefore, area of the ABD = Area of the ACD . Verified
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).
Solution: let , the ratio is
Here, ,
Using section formula , we have
and
Since the line passed through the point P() , then
Therefore, the ratio is 2 : 9 .
2. Find a relation between and if the points (), (1, 2) and (7, 0) are collinear.
Solution: Here, ,
We know that,
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.