NCERT Solutions for Class 10 Maths Unit 7

Coordinate Geometry Class 10

Unit 7 Coordinate Geometry Exercise 7.1, 7.2, 7.3, 7.4 Solutions

"A system of geometry where the position of points on the plane is described using an ordered pair of numbers."

Recall that a plane is a flat surface that goes on forever in both directions. If we were to place a point on the plane, coordinate geometry gives us a way to describe exactly where it is by using two numbers.

Exercise 7.1 : Solutions of Questions on Page Number : 161

Q1 :

Find the distance between the following pairs of points:

(i) (2, 3), (4, 1) (ii) (-5, 7), (-1, 3) (iii) (a, b), (- a, - b)

(i) Distance between the two points is given by (ii) Distance between is given by (iii) Distance between is given by Q2 :

Find the distance between the points (0, 0) and (36, 15). Can you now find the distance between the two towns A and B discussed in Section 7.2.

Distance between points  Yes, we can find the distance between the given towns A and B.

Assume town A at origin point (0, 0).

Therefore, town B will be at point (36, 15) with respect to town A.

And hence, as calculated above, the distance between town A and B will be

39 km.

Q3 :

Determine if the points (1, 5), (2, 3) and (- 2, - 11) are collinear.

Q4 :

Check whether (5, - 2), (6, 4) and (7, - 2) are the vertices of an isosceles triangle.

Q5 :

In a classroom, 4 friends are seated at the points A, B, C and D as shown in the following figure. 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. Q6 :

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)

Q7 :

Find the point on the x-axis which is equidistant from (2, - 5) and (- 2, 9).

Q8 :

Find the values of y for which the distance between the points P (2, - 3) and Q (10, y) is 10 units.

Q9 :

If Q (0, 1) is equidistant from P (5, - 3) and R (x, 6), find the values of x. Also find the distance QR and PR.

Q10 :

If Q (0, 1) is equidistant from P (5, - 3) and R (x, 6), find the values of x. Also find the distance QR and PR.

Q11 :

Find a relation between x and y such that the point (x, y) is equidistant from the point (3, 6) and (- 3, 4).

Q12 :

Find a relation between x and y such that the point (x, y) is equidistant from the point (3, 6) and (- 3, 4).

Exercise 7.2 : Solutions of Questions on Page Number : 167

Q1 :

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

Q2 :

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

Q3 :

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

Q4 :

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

Q5 :

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 flower pots have been placed at a distance of 1 m from each other along AD, as shown in the following figure. Niharika runs the distance AD on the 2nd line and posts a green flag. Preet runs 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? Q6 :

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 flower pots have been placed at a distance of 1 m from each other along AD, as shown in the following figure. Niharika runs the distance AD on the 2nd line and posts a green flag. Preet runs 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? Q7 :

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

Q8 :

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

Q9 :

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.

Q10 :

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.

Q11 :

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

Q12 :

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

Q13 :

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

Q14 :

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

Q15 :

If A and B are ( - 2, - 2) and (2, - 4), respectively, find the coordinates of P such that and P lies on the line segment AB.

Q16 :

If A and B are ( - 2, - 2) and (2, - 4), respectively, find the coordinates of P such that and P lies on the line segment AB.

Q17 :

Find the coordinates of the points which divide the line segment joining A (- 2, 2) and B (2, 8) into four equal parts.

Q18 :

Find the coordinates of the points which divide the line segment joining A (- 2, 2) and B (2, 8) into four equal parts.

Q19 :

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 = (product of its diagonals)]

Q20 :

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 = (product of its diagonals)]

Exercise 7.3 : Solutions of Questions on Page Number : 170

Q1 :

Find the area of the triangle whose vertices are:

(i) (2, 3), (- 1, 0), (2, - 4) (ii) (- 5, - 1), (3, - 5), (5, 2)

Q2 :

Find the area of the triangle whose vertices are:

(i) (2, 3), (- 1, 0), (2, - 4) (ii) (- 5, - 1), (3, - 5), (5, 2)

Q3 :

In each of the following find the value of 'k', for which the points are collinear.

(i) (7, - 2), (5, 1), (3, - k) (ii) (8, 1), (k, - 4), (2, - 5)

Q4 :

In each of the following find the value of 'k', for which the points are collinear.

(i) (7, - 2), (5, 1), (3, - k) (ii) (8, 1), (k, - 4), (2, - 5)

Q5 :

Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, - 1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.

Q6 :

Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, - 1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.

Q7 :

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

Q8 :

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

Q9 :

You have studied in Class IX that a median of a triangle divides it into two triangles of equal areas. Verify this result for ΔABC whose vertices are A (4, - 6), B (3, - 2) and C (5, 2)

Q10 :

You have studied in Class IX that a median of a triangle divides it into two triangles of equal areas. Verify this result for ΔABC whose vertices are A (4, - 6), B (3, - 2) and C (5, 2)

Exercise 7.4 : Solutions of Questions on Page Number : 171

Q1 :

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)

Q2 :

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)

Q3 :

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

Q4 :

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

Q5 :

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

Q6 :

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

Q7 :

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

Q8 :

The class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are 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 following 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 Δ PQR if C is the origin?

Also calculate the areas of the triangles in these cases. What do you observe?

Q9 :

The class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are 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 following 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 Δ PQR if C is the origin?

Also calculate the areas of the triangles in these cases. What do you observe?

Q10 :

The vertices of a ΔABC 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 ΔADE and compare it with the area of ΔABC. (Recall Converse of basic proportionality theorem and Theorem 6.6 related to

ratio of areas of two similar triangles)

Q11 :

The vertices of a ΔABC 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 ΔADE and compare it with the area of ΔABC. (Recall Converse of basic proportionality theorem and Theorem 6.6 related to

ratio of areas of two similar triangles)

Q12 :

Let A (4, 2), B (6, 5) and C (1, 4) be the vertices of ΔABC.

(i) The median from A meets BC at D. Find the coordinates of point D.

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

(iii) Find the coordinates of point 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?

(v) If A(x1, y1), B(x2, y2), and C(x3, y3) are the vertices of ΔABC, find the coordinates of the centroid of the triangle.

Q13 :

Let A (4, 2), B (6, 5) and C (1, 4) be the vertices of ΔABC.

(i) The median from A meets BC at D. Find the coordinates of point D.

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

(iii) Find the coordinates of point 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?

(v) If A(x1, y1), B(x2, y2), and C(x3, y3) are the vertices of ΔABC, find the coordinates of the centroid of the triangle.

Q14 :

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 is a square? a rectangle? or a rhombus? Justify your answer.

Q15 :

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 is a square? a rectangle? or a rhombus? Justify your answer.