# Important Questions for CBSE Class 10 Maths Chapter 7 - Coordinate Geometry 4 Mark Question

## CBSE Class 10 Maths Chapter-7 Coordinate Geometry – Free PDF Download

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CBSE Class 10 Maths Chapter-7 Coordinate Geometry Important Questions

## 4 Mark Questions

1. If the points is equidistant from the points and  prove that
Ans. Let

2. andare three concyclic points whose centre is . Find the possible value of and y.

3. Find the vertices of the triangle, the mid-points of whose sides are and
Ans. Let vertices of  be and
By mid-points formula

Subtracting (i), (iii) and (v) from (vii)
We get,
Subtracting (ii), (iv) and (vi) from eq. (viii)
We get,

4. The two opposite vertices of a square are and. Find the coordinates of the other two vertices.

Ans.

In right

Put the value of y in eq. (i)

Now
And

5. Find the coordinates of the circumcentre of a triangle whose vertices are A(4,6), B(0,4) and C(6,2). Also find its circum-radius.

Ans. Let P be the circum-centre of  then PA = PB = PC

On solving equations (i) and (ii),

6. If two vertices of an equilateral triangle are  Find the third vertex.

Ans.

Put the value of y in eq. (i),

7. If P and Q are two points whose coordinates are and respectively and S is the point show that  is independent of .
Ans.

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

Ans.
= 10.5 sq. units

area of quadrilateral = 10.5 + 17.5 = 28 sq. units.

9. The vertices of  are and . A line is drawn to intersect sides AB and AC at D and E respectively such that . Calculate the area of the and compare it with the area of .

Ans.

Now coordinate of D and E are
and

10. Prove that the points and are the vertices of an equilateraltriangle. Calculate the area of this triangle.
Ans. Let

11. are the vertices of a is the mid-point of BC and P is a point on AD joined such that find the coordinates of P.

Ans. Let and are the vertices of is the mid- point of BC

Coordinate of D
i.e.,
Coordinate of P are

12. The coordinates of the vertices of are. Given that the area of is 12, find the value of K.
Ans.

Value of

13. Find the lengths of the medians of the triangle whose vertices are and .

Ans. Coordinates of points D, E and F are

Length of the median BE

And length of the median CF

14. The area of a triangle is 5. Two of its vertices are and. The third vertex lies on. Find the third vertex.
Ans. Let the third vertex be. Other two vertices of the  are and

lies on eq.
On solving eq.
We get
Similarly, on solving eq.
We get

15. Prove that the point and are collinear, if
Ans. Since  are collinear
Area = 0

Dividing by ,

16. If, Q (0, 1) is equidistant from P (5, –3) and R (x, 6), find the values of x. Also, find the distances QR and PR.
Ans. It is given that Q is equidistant from P and R. UsingDistance Formula, we get
PQ = RQ
PQ2 = RQ2

⇒
⇒
Squaring both sides, we get
⇒ 25+16=x2+25
⇒ x2=16
⇒ x=4,−4
Thus, Q is (4, 6) or (–4, 6).
Using Distance Formula to find QR, we get
Using value of x = 4
QR=

Using value of x = –4
QR=

Therefore, QR=
Using Distance Formula to find PR, we get
Using value of x = 4
PR=

Using value of x =–4
PR=

Therefore, x = 4, –4
QR=, PR=

17. Find the coordinates of the points which divides the line segment joining A(–2, 2) and B(2, 8) into four equal parts.
Ans. A = (–2, 2) and B = (2, 8)
Let P, Q and R are the points which divide line segment AB into 4 equal parts.
Let coordinates of point P =(x1, y1), Q =(x2, y2) and R =(x3, y3)
We know AP = PQ = QR = RS.
It means, point P divides line segment AB in 1:3.
Using Section formula to find coordinates of point P, we get

Since, AP = PQ = QR = RS.
It means, point Q is the mid-point of AB.
Using Section formula to find coordinates of point Q, we get

Because, AP = PQ = QR = RS.
It means, point R divides line segment AB in 3:1
Using Section formula to find coordinates of point P, we get

Therefore, P=(–1, ), Q= (0, )and R =(1, )

18. The two opposite vertices of a square are  and  Find the coordinates of the other two vertices.

Ans. Let ABCD be a square and B be the unknown vertex.
AB = BC
AB2 = BC2

……….(i)
In ABC, AB2 + BC2 = AC2

……….(ii)
Putting the value of  in eq. (ii),

= 0 or 4

19. The class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmoharare planted on the boundary at a distance of 1 m from each other. There is a triangular grassy lawn in the plot as shown in the figure. 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 area of the triangle in these cases. What do you observe?

Ans. (i) Taking A as the origin, AD and AB as the coordinate axes. Clearly, the points P, Q and
R are (4, 6), (3, 2) and (6, 5) respectively.
(ii) Taking C as the origin, CB and CD as the coordinate axes. Clearly, the points P, Q and R are given by (12, 2), (13, 6) and (10, 3) respectively.
We know that the area of the triangle =
Area of PQR (First case) =

=  sq. units
And Area of PQR (Second case) =

=  sq. units
Hence, the areas are same in both the cases.

20. ABCD is a rectangle formed by joining points A B C and D P, Q, R and S are the mid-points of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? Or a rhombus? Justify your answer.
Ans. Using distance formula, PQ =
=
QR =
=
RS =
=
SP =
=
PQ = QR = RS = SP
Now, PR =  =  = 6
And SQ =  = = 5
PR SQ
Since all the sides are equal but the diagonals are not equal.
PQRS is a rhombus.

21. In a classroom, 4 friends are seated at the points A (3, 4), B (6, 7), C (9, 4) and D (6, 1). Champa and Chameli 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.
Ans. We have A = (3, 4), B = (6, 7), C = (9, 4) and D = (6, 1)
Using Distance Formula to find distances AB, BC, CD and DA, we get
AB =

BC =

CD =

DA =

Therefore, All the sides of ABCD are equal here. … (1)
Now, we will check the length of its diagonals.
AC =

BD =

So, Diagonals of ABCD are also equal. … (2)
From (1) and (2), we can definitely say that ABCD is a square.
Therefore, Champa is correct.

22. Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer.
(i) (–1, –2), (1, 0), (–1, 2), (–3, 0)
(ii) (–3, 5), (3, 1), (0, 3) , (–1, –4)
(iii) (4, 5), (7, 6), (4, 3), (1, 2)
Ans. (i) Let A = (–1, –2), B = (1, 0), C= (–1, 2) and D = (–3, 0)
Using Distance Formula to find distances AB, BC, CD and DA, we get
AB =

BC=

CD=

DA=

Therefore, all four sides of quadrilateral are equal. … (1)
Now, we will check the length of diagonals.
AC=

BD=

Therefore, diagonals of quadrilateral ABCD are also equal. … (2)
From (1) and (2), we can say that ABCD is a square.
(ii) Let A = (–3, 5), B= (3, 1), C= (0, 3) and D= (–1, –4)
Using Distance Formula to find distances AB, BC, CD and DA, we get
AB=
BC=
CD=
DA=
We cannot find any relation between the lengths of different sides.
Therefore, we cannot give any name to the quadrilateral ABCD.
(iii) Let A = (4, 5), B= (7, 6), C= (4, 3) and D= (1, 2)
Using Distance Formula to find distances AB, BC, CD and DA, we get
AB=
BC=
CD=
DA=
Here opposite sides of quadrilateral ABCD are equal. … (1)
We can now find out the lengths of diagonals.
AC=
BD=
Here diagonals of ABCD are not equal. … (2)
From (1) and (2), we can say that ABCD is not a rectangle therefore it is a parallelogram.

23. 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 you observe?
(Note: The point which is common to all the three medians is called 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.

Ans. Let A (4, 2), B (6, 5) and C (1, 4) be the vertices of ABC.
(i) Since AD is the median of ABC.
D is the mid-point of BC.
Its coordinates are  =
(ii) Since P divides AD in the ratio 2 : 1
Its coordinates are  =
(iii) Since BE is the median of ABC.
E is the mid-point of AD.
Its coordinates are  =
Since Q divides BE in the ratio 2 : 1.
Its coordinates are  =
Since CF is the median of ABC.
F is the mid-point of AB.
Its coordinates are  =
Since R divides CF in the ratio 2 : 1.
Its coordinates are  =
(iv) We observe that the points P, Q and R coincide, i.e., the medians AD, BE and CF are concurrent at the point . This point is known as the centroid of the triangle.
(v) According to the question, D, E, and F are the mid-points of BC, CA and AB respectively.
Coordinates of D are
Coordinates of a point dividing AD in the ratio 2 : 1 are

The coordinates of E are .
The coordinates of a point dividing BE in the ratio 2 : 1 are

Similarly the coordinates of a point dividing CF in the ratio 2 : 1 are

Thus, the point  is common to AD, BE and CF and divides them in the ratio 2 : 1.
The median of a triangle are concurrent and the coordinates of the centroid are .