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Find the distance between the points:

A(9, 3) and B(15, 11)

A(7, – 4) and B(– 5, 1)

A(– 6, – 4) and B(9, – 12)

A(1, – 3) and B(4, – 6)

P(a + b , a – b) and Q(a – b, a + b)

P(a sin a, acos a) and Q(a cos a, – a sin a)

Find the distance of each of the following points from the origin:

A(5, – 12)

B(– 5, 5)

C(– 4, – 6).

Find all possible values of × for which the distance between the points A(x, – 1) and B(5, 3) is 5 units.

Find all possible values of y for which the distance between the points A(2, – 3) and B(10, y) is 10 units.

Find the values of x for which the distance between the points P(x, 4) and Q(9, 10) is 10 units.

If the point A(x,2) is equidistant from the points B(8, – 2) and C(2, – 2), find the value of x. Also, find the length of AB.

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

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

Find points on the x – axis, each of which is at a distance of 10 units from the point A(11, – 8).

Find the point on the y – axis which is equidistant from the points A(6, 5) and B(– 4, 3).

If the point P(x, y) is equidistant from the points A(5, 1) and B(– 1, 5), prove that 3x = 2y.

If P(x, y) is a point equidistant from the points A(6, – 1) and B(2, 3), show that × – y = 3.

Find the coordinates of the point equidistant from three given points A(5, 3), B(5, – 5) and C(1, – 5).

If the points A(4, 3) and B(x, 5) lie on a circle with the centre O(2, 3), find the value of x.

If the point C(– 2, 3) is equidistant from the points A(3, – 1) and B(x, 8), find the values of x. Also, find the distance BC.

If the point P(2, 2) is equidistant from the points A(– 2, k) and B(– 2k, – 3), find k. Also, find the length of AP.

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

Using the distance formula, show that the given points are collinear:

(i) (1, – 1), (5, 2) and (9, 5)

(ii) (6, 9), (0, 1) and (– 6, – 7)

(iii) (– 1, – 1), (2, 3) and (8, 11)

(iv) (– 2, 5), (0, 1) and (2, – 3).

Show that the points A(7, 10), B(– 2, 5) and C(3, – 4) are the vertices of an isosceles right triangle.

Show that the points A(3, 0), B(6, 4) and C(– 1, 3) are the vertices of an isosceles right triangle.

If A(5, 2), B(2, – 2) and C(– 2, t) are the vertices of a right triangle with ∠B = 90°, then find the value of t.

Prove that the points A(2, 4), B(2, 6) and C(2 + √3, 5) are the vertices of an equilateral triangle.

Show that the points (– 3, – 3), (3, 3) and (– 3√3, 3√3) are the vertices of an equilateral triangle.

Show that the points A(– 5, 6), B(3, 0) and C(9, 8) are the vertices of an isosceles right – angled triangle. Calculate its area.

Show that the points 0(0, 0), A(3, √3) and B(3, – √3) are the vertices of an equilateral triangle. Find the area of this triangle.

Show that the following points are the vertices of a square:

A(3, 2), B(0, 5), C(– 3, 2) and D(0, – 1)

A(6, 2), B(2, 1), C(1, 5) and D(5, 6)

A(0, – 2), B(3, 1), C(0, 4) and D(– 3, 1)

Show that the points A(– 3, 2), B(– 5, – 5), C(2, – 3) and D(4, 4) are the vertices of a rhombus. Find the area of this rhombus. HINT Area of a rhombus = 1/2 × (product of its diagonals).

Show that the points A(3, 0), B(4, 5), C(– 1, 4) and D(– 2, – 1) are the vertices of a rhombus. Find its area.

Show that the points A(6, 1), B(8, 2), C(9, 4) and D(7, 3) are the vertices of a rhombus. Find its area.

Show that the points A(2, 1), B(5, 2), C(6, 4) and D(3, 3) are the angular points of a parallelogram. Is this figure a rectangle?

Show that A(1, 2), B(4, 3), C(6, 6) and D(3, 5) are the vertices of a parallelogram. Show that ABCD is not a rectangle.

Show that the following points are the vertices of a rectangle:

A(– 4, – 1), B(– 2, – 4), C(4, 0) and D(2, 3)

A(2, – 2), B(14, 10), C(11, 13) and D(– 1, 1)

A(0, – 4), B(6, 2), C(3, 5) and D(– 3, – 1)

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

Find the coordinates of the point which divides the join of A(– 5, 11) and B(4, – 7) in the ratio 7 : 2.

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

Point A lies on the line segment PQ joining P(6, – 6) and Q(– 4, – 1) in such a way that . If the point A also lies on the line 3x + k(y + 1) = 0, find the value of k.

Points P, Q, R and S divide the line segment joining the points A(1, 2) and B(6, 7) in five equal parts. Find the coordinates of the points P, Q and R.

Points P, Q and R in that order are dividing a line segment joining A(1, 6) and B(5, – 2) in four equal parts. Find the coordinates of P, Q and R.

The line segment joining the points A(3, – 4) and B(1, 2) is trisected at the points P(p, – 2) and Q(5/3, q). Find the values of p and q.

Find the coordinates of the midpoint of the line segment joining

A(3, 0) and B (– 5, 4)

P(– 11, – 8) and Q(8, – 2).

If (2, p) is the midpoint of the line segment joining the points A(6, – 5) and B(– 2, 11), find the value of p.

The midpoint of the line segment joining A(2a, 4) and B(– 2, 3b) is C(1, 2a + 1). Find the values of a and b.

The line segment joining A(– 2, 9) and B(6, 3) is a diameter of a circle with centre C. Find the coordinates of C.

Find the coordinates of a point A, where AB is a diameter of a circle with centre C(2, – 3) and the other end of the diameter is B(1, 4).

In what ratio does the point P(2, 5) divide the join of A(8, 2) and B(– 6, 9)?

Find the ratio in which the point divides the line segment joining the points and B(2, – 5).

Find 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.

Find the ratio in which the point (– 3, k) divides the join of A(– 5, – 4) and B(– 2, 3). Also, find the value of k. [CBSE 2007]

In what ratio is the line segment joining A(2, – 3) and B(5, 6) divided by the x – axis? Also, find the coordinates of the point of division.

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

In what ratio does the line x – y – 2 = 0 divide the line segment joining the points A (3, –1) and B(8, 9)?

Find the lengths of the medians of a ΔABC whose vertices are A(0, – 1), B(2, 1) and C(0, 3).

Find the centroid of ΔABC whose vertices are A(– 1, 0), B(5, – 2) and C(8, 2)

If G(– 2, 1) is the centroid of a ΔABC and two of its vertices are A(1, – 6) and B(– 5, 2), find the third vertex of the triangle.

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

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

If the points P(a, – 11), Q(5, b), R(2, 15) and S(1, 1) are the vertices of a parallelogram PQRS, find the values of a and b.

If three consecutive vertices of a parallelogram ABCD are A(1, – 2), B(3, 6) and C(5, 10), find its fourth vertex D.

In what ratio does y – axis divide the line segment joining the points (– 4, 7) and (3, – 7)?

If the point lies on the line segment joining the points A(3, – 5) and B(– 7, 9) then find the ratio in which P divides AB. Also, find the value of y.

Find the ratio in which the line segment joining the points A(3, – 3) and B(– 2, 7) is divided by x – axis. Also, find the point of division.

The base QR of an equilateral triangle PQR lies on x – axis. The coordinates of the point Q are (– 4, 0) and origin is the midpoint of the base. Find the coordinates of the points P and R.

The base BC of an equilateral triangle ABC lies on y-axis. The coordinates of point C are (0, –3). The origin is the midpoint of the base. Find the coordinates of the points A and B. Also, find the coordinates of another point D such that ABCD is a rhombus.

Find the ratio in which the point P(– 1, y) lying on the line segment joining points A(– 3, 10) and B(6, – 8) divides it. Also, find the value of y.

ABCD is a rectangle formed by the points A(– 1, – 1), B(– 1, 4), C(5, 4) and D(5, – 1). If P, Q, R and S be the midpoints of AB, BC, CD, and DA respectively, show that PQRS is a rhombus.

The midpoint P of the line segment joining the points A(– 10, 4) and B(– 2, 0) lies on the line segment joining the points C(– 9, – 4) and D(– 4, y). Find the ratio in which P divides CD. Also, find the value of y.

Find the area of ΔABC whose vertices are:

A(1, 2), B(–2, 3) and C(–3, –4)

A(–5, 7), B(–4, –5) and C(4, 5)

A(3, 8), B(–4, 2) and C(5, –1)

A(10, –6), B(2, 5) and C(–1, 3)

Find the area of quadrilateral ABCD whose vertices are A(3, –1), B(9, –5), C(14, 0) and D(9, 19).

Find the area of quadrilateral PQRS whose vertices are P(–5, –3), Q(–4, – 6), R(2, –3) and S(1, 2).

Find the area of quadrilateral ABCD whose vertices are A(–3, –1), B(–2, – 4), C(4, –1) and D(3, 4).

Find the area of quadrilateral ABCD whose vertices are A(–5, 7), B(–4, –5), C(–1, – 6) and D(4, 5).

Find the area of the triangle formed by joining the midpoints of the sides of the triangle whose vertices are A(2, 1), B(4, 3) and C(2, 5).

A(7, –3), B(5, 3) and C(3, –1) are the vertices of a ΔABC and AD is its median. Prove that the median AD divides ΔABC into two triangles of equal areas.

Find the area of LABC with A(1, –4) and midpoints of sides through A being (2, –1) and (0, –1).

A(6, 1), B(8, 2) and C(9, 4) are the vertices of a parallelogram ABCD. If E is the midpoint of DC, find the area of ΔADE.

If the vertices of LABC be A(1, –3), B(4, p) and C(–9, 7) and its area is 15 square units, find the values of p.

Find the value of k so that the area of the triangle with verticesn5 rticles1 A(k + 1,1), B(4, –3) and C(7, –k) is 6 square units.

For what value of k(k > 0) is the area of the triangle with vertices (–2,5) and (k, –4) and (2k + 1,10) equal to 53 square units?

Show that the following points are collinear:

A(2, – 2), B(–3, 8) and C(–1, 4)

A(–5, 1), B(5, 5) and C(10, 7)

A(5, 1), B(1, –1) and C(11, 4)

Find the value of x for which the points A(x, 2), B(–3, –4) and C(7, –5) are collinear.

For what value of x are the points A(–3, 12), B(7, 6) and C(x, 9) collinear?

For what value of y are the points P(1, 4), Q(3, y) and R(–3, 16) are collinear?

Find the value of y for which the points A(–3, 9), B(2, y) and C(4, –5) are collinear.

For what values of k are the points A(8, 1), B( 3, –2k) and C(k, –5) collinear.

Find a relation between x and y, if the points A(2, 1), B(x, y) and C(7, 5) are collinear.

Find a relation between x and y, if the points A(x, y), B(–5, 7) and C(–4, 5) are collinear.

Prove that the points A(a, 0), B(0, b) and C(1, 1) are collinear, if

If the points P(–3, 9), Q(a, b) and R(4, –5) are collinear and a + b = 1, find the values of a and b.

Points A(-1, y) and B(5, 7) lie on a circle with centre O(2, - 3y). Find the values of y.

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

ABCD is a rectangle whose three vertices are B(4, 0), C(4, 3) and D(0, 3). Find the length of one of its diagonal.

If the point P(k -1, 2) is equidistant from the points A(3, k) and B(k, 5), find the values of k.

Find the ratio in which the point P(x, 2) divides the join of A(12, 5) and B(4, -3).

Prove that the diagonals of a rectangle ABCD with vertices A(2, -1), B(5, -1), C(5, 6) and D(2, 6) are equal and bisect each other.

Find the lengths of the medians AD and BE of ΔABC whose vertices are A(7, -3), B(5, 3) and C(3, -1).

If the point C(k, 4) divides the join of A(2, 6) and B(5, 1) in the ratio 2 : 3 then find the value of k.

Find the point on x-axis which is equidistant from points A(-1, 0) and B(5, 0).

Find the distance between the points .

Find the value of a, so that the point (3, a) lies on the line represented by 2x - 3y = 5.

If the points A(4, 3) and B(x, 5) lie on the circle with centre 0(2, 3), find the value of x.

If P(x, y) is equidistant from the points A(7,1) and B(3, 5), find the relation between x and y.

If the centroid of ΔABC having vertices A(a, b), B(b, c) and C(c, a) is the origin, then find the value of (a + b + c).

Find the centroid of ΔABC whose vertices are A(2, 2), B(-4, -4) and C(5, - 8).

In what ratio does the point C(4, 5) divide the join of A(2, 3) and B(7, 8)?

If the points A(2, 3), B(4, k) and C(6, -3) are collinear, find the value of k.

The distance of the point P(-6, 8) from the origin is

The distance of the point (-3, 4) from x-axis is

The point on x-axis which is equidistant from points A(-1, 0) and B(5, 0) is

If R(5, 6) is the midpoint of the line segment AB joining the points A(6, 5) and B(4, y) then y equals

If the point C(k, 4) divides the join of the points A(2, 6) and B(5, 1) in the ratio 2 : 3 then the value of k is

The perimeter of the triangle with vertices (0, 4), (0, 0) and (3, 0) is

If A(1, 3), B(-1, 2), C(2, 5) and D(x, 4) are the vertices of a llgm ABCD then the value of x is

If the points A(x, 2), B(-3, -4) and C(7, -5) are collinear then the value of x is

The area of a triangle with vertices A(5, 0), B(8, 0) and C(8, 4) in square units is

The area of ΔABC with vertices A(a, 0), O(0, 0) and B(0, b) in square units is

If is the midpoint of the line segment joining the points A(-6, 5) and B(-2, 3) then the value of a is

ABCD is a rectangle whose three vertices are B(4, 0), C(4, 3) and D(0, 3). The length of one of its diagonals is

The coordinates of the point P dividing the line segment joining the points A(1, 3) and B(4, 6) in the ratio 2: 1 is

If the coordinates of one end of a diameter of a circle are (2, 3) and the coordinates of its center are (-2, 5), then the coordinates of the other end of the diameter are

In the given figure P(5, -3) and Q(3, y) are the points of trisection of the line segment joining A(7, -2) and B(1, -5). Then, y equals

The midpoint of segment AB is P(0, 4). If the coordinates of B are (-2, 3), then the coordinates of A are

The point P which divides the line segment joining the points A(2, -5) and B(5, 2) in the ratio 2 : 3 lies in the quadrant

If A(-6, 7) and B(-1, -5) are two given points then the distance 2AB is

Which point on the x-axis is equidistant from the points A(7, 6) and B(-3, 4)?

The distance of P(3, 4) from the x-axis is

In what ratio does the x-axis divide the join of A(2, -3) and B(5, 6)?

In what ratio does the y-axis divide the join of P(-4, 2) and Q(8, 3)?

If P(-1, 1) is the midpoint of the line segment joining A(-3, b) and B(1, b + 4) then b = ?

The line 2x + y - 4 = 0 divides the line segment joining A(2, -2) and B(3, 7) in the ratio

If A(4, 2), B(6, 5) and C(1, 4) be the vertices of ∆ABC and AD is a median, then the coordinates of D are

If A(-1, 0), B(5, -2) and C(8, 2) are the vertices of a ∆ABC then its centroid is

Two vertices of ABC are A (-1, 4) and B(5, 2) and its centroid is G(0, -3). Then, the coordinates of C are

The points A(-4, 0), B(4, 0) and C(0, 3) are the vertices of a triangle, which is

The points P(0, 6), Q(-5, 3) and R(3, 1) are the vertices of a triangle, which is

If the points A(2, 3), B(5, k) and C(6, 7) are collinear then

If the points A(1. 2), O(0, 0) and C(a, b) are collinear then

The area of ΔABC with vertices A(3, 0), B(7, 0) and C(8, 4) is

AOBC is a rectangle whose the vertices are A(0, 3), O(0, 0) and B(5, 0). The length of each of its diagonals is

If the distance between the point A(4, p) and B(1, 0) is 5 then