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# R.D. Sharma solutions for Class 10th Board Exam Mathematics chapter 14 - Co-Ordinate Geometry

## Chapter 14 - Co-Ordinate Geometry

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On which axis do the following points lie?

P(5, 0)

On which axis do the following points lie?

P(5, 0)

On which axis do the following points lie?

Q(0, -2)

On which axis do the following points lie?

Q(0, -2)

On which axis do the following points lie?

R(−4,0)

On which axis do the following points lie?

R(−4,0)

On which axis do the following points lie?

S(0,5)

On which axis do the following points lie?

S(0,5)

Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when A coincides with the origin and AB and AD are along OX and OY respectively.

Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when A coincides with the origin and AB and AD are along OX and OY respectively.

Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when The centre of the square is at the origin and coordinate axes are parallel to the sides AB and AD respectively.

Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when The centre of the square is at the origin and coordinate axes are parallel to the sides AB and AD respectively.

The base PQ of two equilateral triangles PQR and PQR' with side 2a lies along y-axis such that the mid-point of PQ is at the origin. Find the coordinates of the vertices R and R' of the triangles.

The base PQ of two equilateral triangles PQR and PQR' with side 2a lies along y-axis such that the mid-point of PQ is at the origin. Find the coordinates of the vertices R and R' of the triangles.

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Find the distance between the following pair of points:

(-6, 7) and (-1, -5)

Find the distance between the following pair of points:

(a+b, b+c) and (a-b, c-b)

Find the distance between the following pair of points:

(asinα, −bcosα) and (−acos α, bsin α)

Find the distance between the following pair of points:

(a, 0) and (0, b)

Find the distance between the following pair of points:

(a, 0) and (0, b)

Find the value of a when the distance between the points (3, a) and (4, 1) is sqrt10

If the points (2, 1) and (1, -2) are equidistant from the point (xy), show that x + 3y = 0.

Find the values of x, y if the distances of the point (xy) from (-3, 0)  as well as from (3, 0) are 4.

The length of a line segment is of 10 units and the coordinates of one end-point are (2, -3). If the abscissa of the other end is 10, find the ordinate of the other end.

Show that the points A (1, −2), B (3, 6), C (5, 10) and D (3, 2) are the vertices of a parallelogram.

Prove that the points A(1, 7), B (4, 2), C(−1, −1) D (−4, 4) are the vertices of a square.

Prove that the points (3, 0), (6, 4) and (-1, 3) are the vertices of a right-angled isosceles triangle.

Prove that the points (3, 0), (6, 4) and (-1, 3) are the vertices of a right-angled isosceles triangle.

Prove that (2, -2) (-2, 1) and (5, 2) are the vertices of a right-angled triangle. Find the area of the triangle and the length of the hypotenuse.

Prove that the points (2a, 4a), (2a, 6a) and (2a + sqrt3a, 5a) are the vertices of an equilateral triangle.

Prove that the points (2,3), (-4, -6) and (1, 3/2) do not form a triangle.

An equilateral triangle has two vertices at the points (3, 4) and (−2, 3), find the coordinates of the third vertex.

Show that the quadrilateral whose vertices are (2, −1), (3, 4) (−2, 3) and (−3,−2) is a rhombus.

Two vertices of an isosceles triangle are (2, 0) and (2, 5). Find the third vertex if the length of the equal sides is 3.

Two vertices of an isosceles triangle are (2, 0) and (2, 5). Find the third vertex if the length of the equal sides is 3.

Two vertices of an isosceles triangle are (2, 0) and (2, 5). Find the third vertex if the length of the equal sides is 3.

Two vertices of an isosceles triangle are (2, 0) and (2, 5). Find the third vertex if the length of the equal sides is 3.

Which point on the x-axis is equidistant from (5, 9) and (−4, 6)?

Which point on the x-axis is equidistant from (5, 9) and (−4, 6)?

Prove that the points (−2, 5), (0, 1) and (2, −3)  are collinear.

Prove that the points (−2, 5), (0, 1) and (2, −3)  are collinear.

The coordinates of the point P are (−3, 2). Find the coordinates of the point Q which lies on the line joining P and origin such that OP = OQ.

The coordinates of the point P are (−3, 2). Find the coordinates of the point Q which lies on the line joining P and origin such that OP = OQ.

Which point on the y-axis is equidistant from (2, 3)  and (−4, 1)?

Which point on the y-axis is equidistant from (2, 3)  and (−4, 1)?

The three vertices of a parallelogram are (3, 4) (3, 8) and (9, 8). Find the fourth vertex.

The three vertices of a parallelogram are (3, 4) (3, 8) and (9, 8). Find the fourth vertex.

Find the circumcenter of the triangle whose vertices are (-2, -3), (-1, 0), (7, -6).

Find the angle subtended at the origin by the line segment whose end points are (0, 100) and (10, 0).

Find the centre of the circle passing through (5, -8), (2, -9) and (2, 1).

Find the centre of the circle passing through (5, -8), (2, -9) and (2, 1).

Find the value of k, if the point P (0, 2) is equidistant from (3, k) and (k, 5).

Find the value of k, if the point P (0, 2) is equidistant from (3, k) and (k, 5).

If two opposite vertices of a square are (5, 4) and (1, −6), find the coordinates of its remaining two vertices.

If two opposite vertices of a square are (5, 4) and (1, −6), find the coordinates of its remaining two vertices.

Show that A(-3, 2), B(-5,-5), C(2,-3) and D(4,4) are the vertices of a rhombus.A

Show that A(-3, 2), B(-5,-5), C(2,-3) and D(4,4) are the vertices of a rhombus.A

Find the coordinates of the circumcentre of the triangle whose vertices are (3, 0), (-1, -6) and (4, -1). Also, find its circumradius.

Find the coordinates of the circumcentre of the triangle whose vertices are (3, 0), (-1, -6) and (4, -1). Also, find its circumradius.

Find a point on the x-axis which is equidistant from the points (7, 6) and (−3, 4).

Find a point on the x-axis which is equidistant from the points (7, 6) and (−3, 4).

Show that the points A(5, 6), B(1, 5), C(2, 1) and D(6,2) are the vertices of a square.

Show that the points A(5, 6), B(1, 5), C(2, 1) and D(6,2) are the vertices of a square.

Find the point on x-axis which is equidistant from the points (−2, 5) and (2,−3).

Find the point on x-axis which is equidistant from the points (−2, 5) and (2,−3).

Find the value of x such that PQ = QR where the coordinates of P, Q and R are (6, -1), (1, 3) and (x, 8) respectively.

Find the value of x such that PQ = QR where the coordinates of P, Q and R are (6, -1), (1, 3) and (x, 8) respectively.

Prove that the points (0, 0), (5, 5) and (-5, 5) are the vertices of a right isosceles triangle.

Prove that the points (0, 0), (5, 5) and (-5, 5) are the vertices of a right isosceles triangle.

If the point P(x, y ) is equidistant from the points A(5, 1) and B (1, 5), prove that x = y.

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

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

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

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

A(-1,-2) B(1, 0), C (-1, 2), D(-3, 0)

A(-1,-2) B(1, 0), C (-1, 2), D(-3, 0)

A(-3, 5) B(3, 1), C (0, 3), D(-1, -4)

A(-3, 5) B(3, 1), C (0, 3), D(-1, -4)

A(4, 5) B(7, 6), C (4, 3), D(1, 2)

A(4, 5) B(7, 6), C (4, 3), D(1, 2)

Find the equation of the perpendicular bisector of the line segment joining points (7, 1) and (3,5).

Find the equation of the perpendicular bisector of the line segment joining points (7, 1) and (3,5).

Prove that the points (3, 0), (4, 5), (-1, 4) and (-2, -1), taken in order, form a rhombus.
Also, find its area.

Prove that the points (3, 0), (4, 5), (-1, 4) and (-2, -1), taken in order, form a rhombus.
Also, find its area.

In the seating arrangement of desks in a classroom three students Rohini, Sandhya and Bina are seated at A(3, 1), B(6, 4), and C(8, 6). Do you think they are seated in a line?

In the seating arrangement of desks in a classroom three students Rohini, Sandhya and Bina are seated at A(3, 1), B(6, 4), and C(8, 6). Do you think they are seated in a line?

Find a point on y-axis which is equidistant from the points (5, -2) and (-3, 2).

Find a point on y-axis which is equidistant from the points (5, -2) and (-3, 2).

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

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

If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p. Also, find the length of AB.

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Find the coordinates of the point which divides the line segment joining (−1,3) and (4, −7) internally in the ratio 3 : 4

Find the coordinates of the point which divides the line segment joining (−1,3) and (4, −7) internally in the ratio 3 : 4

Find the points of trisection of the line segment joining the points:

5, −6 and (−7, 5),

Find the points of trisection of the line segment joining the points:

5, −6 and (−7, 5),

Find the points of trisection of the line segment joining the points:

(3, -2) and (-3, -4)

Find the points of trisection of the line segment joining the points:

(3, -2) and (-3, -4)

Find the points of trisection of the line segment joining the points:

(2, -2) and (-7, 4).

Find the points of trisection of the line segment joining the points:

(2, -2) and (-7, 4).

Find the coordinates of the point where the diagonals of the parallelogram formed by joining the points (-2, -1), (1, 0), (4, 3) and(1, 2) meet

Find the coordinates of the point where the diagonals of the parallelogram formed by joining the points (-2, -1), (1, 0), (4, 3) and(1, 2) meet

Prove that the points (3, -2), (4, 0), (6, -3) and (5, -5) are the vertices of a parallelogram.

Prove that the points (3, -2), (4, 0), (6, -3) and (5, -5) are the vertices of a parallelogram.

Three consecutive vertices of a parallelogram are (-2,-1), (1, 0) and (4, 3). Find the fourth vertex.

Three consecutive vertices of a parallelogram are (-2,-1), (1, 0) and (4, 3). Find the fourth vertex.

The points (3, -4) and (-6, 2) are the extremities of a diagonal of a parallelogram. If the third vertex is (-1, -3). Find the coordinates of the fourth vertex.

The points (3, -4) and (-6, 2) are the extremities of a diagonal of a parallelogram. If the third vertex is (-1, -3). Find the coordinates of the fourth vertex.

Find the ratio in which the point (2, y) divides the line segment joining the points A (-2,2) and B (3, 7). Also, find the value of y.

Find the ratio in which the point (2, y) divides the line segment joining the points A (-2,2) and B (3, 7). Also, find the value of y.

If A (-1, 3), B (1, -1) and C (5, 1) are the vertices of a triangle ABC, find the length of the median through A.

If the coordinates of the mid-points of the sides of a triangle are (1, 1), (2, —3) and (3, 4), find the vertices of the triangle.

If a vertex of a triangle be (1, 1) and the middle points of the sides through it be (-2,-3) and (5 2) find the other vertices.

In what ratio is the line segment joining the points (-2,-3) and (3, 7) divided by the y-axis? Also, find the coordinates of the point of division.

In what ratio is the line segment joining the points (-2,-3) and (3, 7) divided by the y-axis? Also, find the coordinates of the point of division.

In what ratio is the line segment joining (-3, -1) and (-8, -9) divided at the point (-5, -21/5)?

In what ratio is the line segment joining (-3, -1) and (-8, -9) divided at the point (-5, -21/5)?

If the mid-point of the line joining (3, 4) and (k, 7) is (x, y) and 2x + 2y + 1 = 0 find the value of k.

Determine the ratio in which the straight line x - y - 2 = 0 divides the line segment
joining (3, -1) and (8, 9).

Determine the ratio in which the straight line x - y - 2 = 0 divides the line segment
joining (3, -1) and (8, 9).

Find the ratio in which the line segment joining (-2, -3) and (5, 6) is divided by x-axis Also, find the coordinates of the point of division in each case.

Find the ratio in which the line segment joining (-2, -3) and (5, 6) is divided by x-axis Also, find the coordinates of the point of division in each case.

Find the ratio in which the line segment joining (-2, -3) and (5, 6) is divided by y-axis. Also, find the coordinates of the point of division in each case.

Find the ratio in which the line segment joining (-2, -3) and (5, 6) is divided by y-axis. Also, find the coordinates of the point of division in each case.

Prove that the points (4, 5) (7, 6), (6, 3) (3, 2) are the vertices of a parallelogram. Is it a rectangle.

Prove that the points (4, 5) (7, 6), (6, 3) (3, 2) are the vertices of a parallelogram. Is it a rectangle.

Prove that (4, 3), (6, 4) (5, 6) and (3, 5)  are the angular points of a square.

Prove that (4, 3), (6, 4) (5, 6) and (3, 5)  are the angular points of a square.

Prove that the points A(-4,-1), B(-2, 4), C(4, 0) and D(2, 3) are the vertices of a rectangle.

Prove that the points A(-4,-1), B(-2, 4), C(4, 0) and D(2, 3) are the vertices of a rectangle.

Find the lengths of the medians of a triangle whose vertices are A (−1,3), B(1,−1) and C(5, 1).

Three vertices of a parallelogram are (a+b, a-b), (2a+b, 2a-b), (a-b, a+b). Find the fourth vertex.

If two vertices of a parallelogram are (3, 2) (-1, 0) and the diagonals cut at (2, -5), find the other vertices of the parallelogram.

If the coordinates of the mid-points of the sides of a triangle are (3, 4) (4, 6) and (5, 7), find its vertices.

The line segment joining the points P(3, 3) and Q(6, -6) is trisected at the points A and B such that Ais nearer to P. If A also lies on the line given by 2x + y + k = 0, find the value of k.

The line segment joining the points P(3, 3) and Q(6, -6) is trisected at the points A and B such that Ais nearer to P. If A also lies on the line given by 2x + y + k = 0, find the value of k.

If the points (-2, -1), (1, 0), (x, 3) and  (1, y) form a parallelogram, find the values of x and y.

The points A(2, 0), B(9, 1) C(11, 6) and D(4, 4) are the vertices of a quadrilateral ABCD. Determine whether ABCD is a rhombus or not.

The points A(2, 0), B(9, 1) C(11, 6) and D(4, 4) are the vertices of a quadrilateral ABCD. Determine whether ABCD is a rhombus or not.

If three consecutive vertices of a parallelogram are (1, -2), (3, 6) and (5, 10), find its fourth vertex.

If three consecutive vertices of a parallelogram are (1, -2), (3, 6) and (5, 10), find its fourth vertex.

If the points A (a, -11), B (5, b), C (2, 15) and D (1, 1) are the vertices of a parallelogram ABCD, find the values of a and b.

If the points A (a, -11), B (5, b), C (2, 15) and D (1, 1) are the vertices of a parallelogram ABCD, find the values of a and b.

If the coordinates of the mid-points of the sides of a triangle be (3, -2), (-3, 1) and (4, -3), then find the coordinates of its vertices.

If the coordinates of the mid-points of the sides of a triangle be (3, -2), (-3, 1) and (4, -3), then find the coordinates of its vertices.

Find the length of the medians of a ΔABC having vertices at A(0, -1), B(2, 1) and C(0, 3).

Find the lengths of the medians of a ΔABC having vertices at A(5, 1), B(1, 5), and C(-3, -1).

Find the coordinates of the points which divide the line segment joining the points (-4, 0) and (0, 6) in four equal parts.

Find the coordinates of the points which divide the line segment joining the points (-4, 0) and (0, 6) in four equal parts.

Show that the mid-point of the line segment joining the points (5, 7) and (3, 9) is also the mid-point of the line segment joining the points (8, 6) and (0, 10).

Find the distance of the point (1, 2) from the mid-point of the line segment joining the points (6, 8) and (2, 4).

If A and B are (1, 4) and (5, 2) respectively, find the coordinates of P when AP/BP = 3/4.

If A and B are (1, 4) and (5, 2) respectively, find the coordinates of P when AP/BP = 3/4.

Show that the points A (1, 0), B (5, 3), C (2, 7) and D (−2, 4) are the vertices of a parallelogram.

Show that the points A (1, 0), B (5, 3), C (2, 7) and D (−2, 4) are the vertices of a parallelogram.

Determine the ratio in which the point P (m, 6) divides the join of A(-4, 3) and B(2, 8). Also, find the value of m.

Determine the ratio in which the point P (m, 6) divides the join of A(-4, 3) and B(2, 8). Also, find the value of m.

Determine the ratio in which the point (-6, a) divides the join of A (-3, 1)  and B (-8, 9). Also, find the value of a.

Determine the ratio in which the point (-6, a) divides the join of A (-3, 1)  and B (-8, 9). Also, find the value of a.

The line segment joining the points (3, -4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, -2) and (5/3, q) respectively. Find the values of p and q.

The line joining the points (2, 1) and (5, -8) is trisected at the points P and Q. If point Plies on the line 2x - y + k = 0. Find the value of k.

The line joining the points (2, 1) and (5, -8) is trisected at the points P and Q. If point Plies on the line 2x - y + k = 0. Find the value of k.

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.

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

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.

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.

Find the ratio in which y-axis divides the line segment joining the points A(5, –6) and B(–1, –4). Also find the coordinates of the point of division.

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

In what ratio does the point (−4, 6) divide the line segment joining the points A(−6, 10) and B(3,−8)?

In what ratio does the point (−4, 6) divide the line segment joining the points A(−6, 10) and B(3,−8)?

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

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

A point P divides the line segment joining the points A(3, -5) and B(-4, 8) such that (AP)/(PB) = k/1. If P lies on the line x + y = 0, then find the value of k.

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Find the centroid of the triangle whosw vertices are:

(1) (1,4), (-1,1) and (3,2)

Two vertices of a triangle are (1, 2), (3, 5) and its centroid is at the origin. Find the coordinates of the third vertex.

Prove analytically that the line segment joining the middle points of two sides of a triangle is equal to half of the third side.

Prove that the lines joining the middle points of the opposite sides of a quadrilateral and the join of the middle points of its diagonals meet in a point and bisect one another

If G be the centroid of a triangle ABC and P be any other point in the plane, prove that PA2+ PB2 + PC2 = GA2 + GB2 + GC2 + 3GP2.

If G be the centroid of a triangle ABC, prove that:

AB2 + BC2 + CA2 = 3 (GA2 + GB2 + GC2)

If G be the centroid of a triangle ABC, prove that:

AB2 + BC2 + CA2 = 3 (GA2 + GB2 + GC2)

If (−2, 3), (4, −3) and (4, 5) are the mid-points of the sides of a triangle, find the coordinates of its centroid.

If (−2, 3), (4, −3) and (4, 5) are the mid-points of the sides of a triangle, find the coordinates of its centroid.

In Fig. 14.36, a right triangle BOA is given C is the mid-point of the hypotenuse AB. Show that it is equidistant from the vertices O, A  and B.

We have a right angled triangle,triangle BOA  right angled at O. Co-ordinates are B (0,2b); A (2a0) and C (0, 0).

In Fig. 14.36, a right triangle BOA is given C is the mid-point of the hypotenuse AB. Show that it is equidistant from the vertices O, A  and B.

We have a right angled triangle,triangle BOA  right angled at O. Co-ordinates are B (0,2b); A (2a0) and C (0, 0).

Find the third vertex of a triangle, if two of its vertices are at (−3, 1) and (0, −2) and the centroid is at the origin.

Find the third vertex of a triangle, if two of its vertices are at (−3, 1) and (0, −2) and the centroid is at the origin.

A (3, 2) and B (−2, 1)  are two vertices of a triangle ABC whose centroid G has the coordinates (5/3,-1/3)Find the coordinates of the third vertex C of the triangle.

A (3, 2) and B (−2, 1)  are two vertices of a triangle ABC whose centroid G has the coordinates (5/3,-1/3)Find the coordinates of the third vertex C of the triangle.

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Find the area of a triangle whose vertices are

(6,3), (-3,5) and (4,2)

Find the area of a triangle whose vertices are

(at_1^2,2at_1),(at_2^2,2at_2) and (at_3^2,2at_3)

Find the area of a triangle whose vertices are

(a, c + a), (a, c) and (−a, c − a)

Find the area of the quadrilaterals, the coordinates of whose vertices are

(−3, 2), (5, 4), (7, −6) and (−5, −4)

Find the area of the quadrilaterals, the coordinates of whose vertices are

(1, 2), (6, 2), (5, 3) and (3, 4)

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

The four vertices of a quadrilateral are (1, 2), (−5, 6), (7, −4) and (k, −2) taken in order. If the area of the quadrilateral is zero, find the value of k.

The vertices of ΔABC are (−2, 1), (5, 4)  and (2, −3)  respectively. Find the area of the triangle and the length of the altitude through A.

Show that the following sets of points are collinear.

(2, 5), (4, 6) and (8, 8)

Show that the following sets of points are collinear.

(1, −1), (2, 1) and (4, 5)

Prove that the points (a, 0), (0, b) and (1, 1) are collinear if 1/a+1/b=1

The point A divides the join of P (−5, 1)  and Q(3, 5) in the ratio k:1. Find the two values of kfor which the area of ΔABC where B is (1, 5) and C(7, −2) is equal to 2 units.

The area of a triangle is 5. Two of its vertices are (2, 1) and (3, −2). The third vertex lies on y = x + 3. Find the third vertex.

If a≠ b ≠ c, prove that the points (a, a2), (bb2), (cc2) can never be collinear.

Four points A (6, 3), B (−3, 5), C(4, −2) and D (x, 3x) are given in such a way that (ΔDBG) /(ΔABG)=1/2, find x

For what value of a point (a, 1), (1, -1) and (11, 4) are collinear?

Prove that the points (a, b), (a1, b1) and (a −a1, b −b1) are collinear if ab1 = a1b.

## R.D. Sharma solutions for Class 10th Board Exam Mathematics chapter 14 - Co-Ordinate Geometry

R.D. Sharma solutions for Class 10th Board Exam Mathematics chapter 14 (Co-Ordinate Geometry) include all questions with solution and detail explanation from 10 Mathematics. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has created the CBSE 10 Mathematics solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 10th Board Exam Mathematics chapter 14 Co-Ordinate Geometry are Area of a Triangle, Section Formula, Distance Formula, Graphs of Linear Equations, Concepts of Coordinate Geometry, Coordinate Geometry Examples and Solutions, Slope of a Line, Intercepts Made by a Line, Standard Forms of Equation of a Line, General Equation of a Line, Concepts of Coordinate Geometry.

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