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# NCERT solutions Mathematics Class 10 chapter 7 Coordinate Geometry

## Chapter 7 - Coordinate Geometry

#### Pages 161 - 162

Find the distance between the following pair of point (2, 3), (4, 1)

Q 1.1 | Page 161

Find the distance between the following pairs of points: (−5, 7), (−1, 3)

Q 1.2 | Page 161

Find the distance between the following pair of point:(ab), (− a, − b)

Q 1.3 | Page 161

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.

Q 2 | Page 161

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

Q 4 | Page 161

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.

Q 5 | Page 161

Name the type of quadrilateral formed, if any, by the given points, and give reasons for your answer: (- 1, - 2), (1, 0), (- 1, 2), (- 3, 0)

Q 6.1 | Page 161

Name the type of quadrilateral formed, if any, by the following point, and give reasons for your answer: (− 3, 5), (3, 1), (0, 3), (− 1, − 4)

Q 6.2 | Page 161

Name the type of quadrilateral formed, if any, by the following point, and give reasons for your answer (4, 5), (7, 6), (4, 3), (1, 2)

Q 6.3 | Page 161

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

Q 7 | Page 161

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

Q 8 | Page 161

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.

Q 9 | Page 162

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

Q 10 | Page 162

#### Page 167

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

Q 1 | Page 167

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

Q 2 | Page 167

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 1/4 th the distance AD on the 2nd line and posts a green flag. Preet runs 1/5

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 joining the two flags, where should she post her flag?

Q 3 | Page 167

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.

Q 5 | Page 167

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

Q 6 | Page 167

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

Q 7 | Page 167

If A and B are (− 2, − 2) and (2, − 4), respectively, find the coordinates of P such that AP = 3/7 AB and P lies on the line segment AB.

Q 8 | Page 167

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

Q 9 | Page 167

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 = 1/2 (product of its diagonals)]

Q 10 | Page 167

#### Page 170

Find the area of the triangle whose vertices are: (2, 3), (-1, 0), (2, -4)

Q 1.1 | Page 170

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

(7, -2), (5, 1), (3, -k

Q 2.1 | Page 170

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

(8, 1), (k, -4), (2, -5)

Q 2.2 | Page 170

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

Q 3 | Page 170

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

Q 4 | Page 170

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).

Q 5 | Page 170

#### Page 171

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).

Q 1 | Page 171

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

Q 2 | Page 171

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

Q 3 | Page 171

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

Q 4 | Page 171

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?

Q 5 | Page 171

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 (AD)/(AB) = (AE)/(AC) = 1/4Calculate 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)

Q 6 | Page 171

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(x1y1), B(x2y2), and C(x3y3) are the vertices of ΔABC, find the coordinates of the centroid of the triangle.

Q 7 | Page 171

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 midpoints of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.

Q 8 | Page 171

#### Extra questions

If the opposite vertices of a square are (1, – 1) and (3, 4), find the coordinates of the remaining angular points.

Find the area of a triangle whose vertices are A(3, 2), B (11, 8) and C(8, 12).

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

If D, E and F are the mid-points of sides BC, CA and AB respectively of a ∆ABC, then using coordinate geometry prove that Area of ∆DEF = \frac { 1 }{ 4 } "(Area of ∆ABC)"

If A(4, –6), B(3, –2) and C(5, 2) are the vertices of ∆ABC, then verify the fact that a median of a triangle ABC divides it into two triangle of equal areas.

The coordinates of A, B, C are (6, 3), (–3, 5) and (4, – 2) respectively and P is any point (x, y). Show that the ratio of the areas of triangle PBC and ABC is

For what value of k are the points (k, 2 – 2k), (–k + 1, 2k) and (–4 – k, 6 – 2k) are collinear ?

Prove that the points (2, – 2), (–3, 8) and (–1, 4) are collinear

Find the area of the quadrilateral ABCD whose vertices are respectively A(1, 1), B(7, –3), C(12, 2) and D(7, 21).

Prove that the area of a triangle with vertices (t, t −2), (t + 2, t + 2) and (t + 3, t) is independent of t.

Prove that the points (a, b + c), (b, c + a) and (c, a + b) are collinear

If the coordinates of two points A and B are (3, 4) and (5, – 2) respectively. Find the coordniates of any point P, if PA = PB and Area of ∆PAB = 10

For what value of x will the points (x, –1), (2, 1) and (4, 5) lie on a line ?

The vertices of ∆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 \frac{AD}{AB}=\frac{AE}{AC}=\frac{1}{4} .Calculate the area of ∆ADE and compare it with the area of ∆ABC

If the coordinates of the mid points of the sides of a triangle are (1, 1), (2, – 3) and (3, 4) Find its centroid

Show that four points (0, – 1), (6, 7), (–2, 3) and (8, 3) are the vertices of a rectangle. Also, find its area

Find the coordinates of the circumcentre of the triangle whose vertices are (8, 6), (8, – 2) and (2, – 2). Also, find its circum radius

Find the distance between two points

(i) P(–6, 7) and Q(–1, –5)

(ii) R(a + b, a – b) and S(a – b, –a – b)

(iii) A(at_1^2,2at_1)" and " B(at_2^2,2at_2)

If the point P(x, y) is equidistant from the points A(a + b, b – a) and B(a – b, a + b). Prove that bx = ay.

If two vertices of an equilateral triangle be (0, 0), (3, √3 ), find the third vertex

Show that the points (a, a), (–a, –a) and (– √3 a, √3 a) are the vertices of an equilateral triangle. Also find its area.

Find the coordinates of the centre of the circle passing through the points (0, 0), (–2, 1) and (–3, 2). Also, find its radius.

If P and Q are two points whose coordinates are (at2 ,2at) and (a/t2 , 2a/t) respectively and S is the point (a, 0). Show that \frac{1}{SP}+\frac{1}{SQ} is independent of t.

Find the value of x, if the distance between the points (x, – 1) and (3, 2) is 5.

If P (2, – 1), Q(3, 4), R(–2, 3) and S(–3, –2) be four points in a plane, show that PQRS is a rhombus but not a square. Find the area of the rhombus

Prove that the points (–3, 0), (1, –3) and (4, 1) are the vertices of an isosceles right angled triangle. Find the area of this triangle

Show that the points (1, – 1), (5, 2) and (9, 5) are collinear.

The three vertices of a parallelogram taken in order are (–1, 0), (3, 1) and (2, 2) respectively. Find the coordinates of the fourth vertex.

Find the coordinates of the point which divides the line segment joining the points (6, 3) and (– 4, 5) in the ratio 3 : 2 internally.

If A(–2, –1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram, find the values of a and b

Prove that (4, – 1), (6, 0), (7, 2) and (5, 1) are the vertices of a rhombus. Is it a square?

Two vertices of a triangle are (3, –5) and (–7, 4). If its centroid is (2, –1). Find the third vertex

Find the area of the triangle whose vertices are: (8, 1), (k, -4), (2, -5)

If the points A (6, 1), B (8, 2), C(9, 4) and D(p, 3) are vertices of a parallelogram, taken in order, find the value of p

Find the coordinates of the centroid of a triangle whose vertices are (–1, 0), (5, –2) and (8, 2)

Find the lengths of the medians of a ∆ABC whose vertices are A(7, –3), B(5,3) and C(3,–1)

In what ratio does the x-axis divide the line segment joining the points (2, –3) and (5, 6)? Also, find the coordinates of the point of intersection.

Determine the ratio in which the line 3x + y – 9 = 0 divides the segment joining the points (1, 3) and (2, 7)

If A (5, –1), B(–3, –2) and C(–1, 8) are the vertices of triangle ABC, find the length of median through A and the coordinates of the centroid.

If the point C (–1, 2) divides internally the line segment joining A (2, 5) and B in ratio 3 : 4, find the coordinates of B

Prove that the points (–2, –1), (1, 0), (4, 3) and (1, 2) are the vertices of a parallelogram. Is it a rectangle ?

Prove that the diagonals of a rectangle bisect each other and are equal.

If the coordinates of the mid-points of the sides of a triangle are (1, 2) (0, –1) and (2, 1). Find the coordinates of its vertices.

Find the coordinates of points which trisect the line segment joining (1, –2) and (–3, 4)

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