The coordinates of the point C dividing the line segment joining the points P(2, 4) and Q(5, 7) internally in the ratio 2:1 is______.

If \(P(\frac{a}{3},\frac{b}{2})\)is the mid-point of the line segment joining A(−4, 3) and B(−2, 4) then (a, b) is ______.

In what ratio does the point Q(1, 6) divide the line segment joining the points P(2, 7) and R(−2, 3) ______.

The ratio in which the x-axis divides the line segment joining the points A(a₁, b₁) and B(a₂, b₂ ) is ______.

The ratio in which the x-axis divides the line segment joining the points (6, 4) and (1, −7) is ______.

If the coordinates of the mid-points of the sides AB, BC and CA of a triangle are (3, 4), (1, 1) and (2, −3) respectively, then the vertices A and B of the triangle are ______.

The mid-point of the line joining (−a, 2b) and (−3a,−4b) is ______.

In what ratio does the y-axis divides the line joining the points (−5, 1) and (2, 3) internally ______.

If (1,−2), (3, 6), (x, 10) and (3, 2) are the vertices of the parallelogram taken in order, then the value of x is ______.

The centroid of the triangle with vertices (−1, −6), (−2, 12) and (9, 3) is

The centre of a circle is (−4, 2). If one end of the diameter of the circle is (−3, 7) then find the other end.

The mid-point of the sides of a triangle are (2, 4), (−2, 3) and (5, 2). Find the coordinates of the vertices of the triangle.

O(0,0) is the centre of a circle whose one chord is AB, where the points A and B are (8,6) and (10,0) respectively. OD is the perpendicular from the centre to the chord AB. Find the coordinates of the mid-point of OD.

The points A(−5, 4) , B(−1, −2) and C(5,2) are the vertices of an isosceles rightangled triangle where the right angle is at B. Find the coordinates of D so that ABCD is a square.

Show that the line segment joining the mid-points of two sides of a triangle is half of the third side
(Hint: Place triangle ABC in a clever way such that A is (0, 0), B is (2a, 0) and C to be (2b, 2c). Now consider the line segment joining the mid-points of AC and BC. This will make calculations simpler).

Prove that the diagonals of the parallellogram bisect each other. [Hint: Take scale on both axes as 1 cm = a units]

Find the coordinates of the point which divides the line segment joining A(−5,11) and B(4,−7) in the ratio 7:2.

The line segment joining A(6, 3) and B(−1, −4) is doubled in length by adding half of AB to each end. Find the coordinates of the new end points.

A line segment AB is increased along its length by 25% by producing it to C on the side of B. If A and B have the coordinates (−2,−3) and (2,1) respectively, then find the coordinates of C.

A car travels at an uniform speed. At 2 pm it is at a distance of 180 km and at 6pm it is at 360 km. Using section formula, find at what distance it will reach 12 midnight.

If the centroid of a triangle is at (4, −2) and two of its vertices are (3, −2) and (5, 2) then find the third vertex of the triangle.

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

The vertices of a triangle are (1, 2), (h, −3) and (−4, k). If the centroid of the triangle is at the point (5, −1) then find the value of \(\sqrt { { (h+k) }^{ 2 }+{ (h+3k) }^{ 2 } } \)

ABC is a triangle whose vertices are A(3, 4), B(−2, −1) and C(5, 3) . If G is the centroid and BDCG is a parallelogram then find the coordinates of the vertex D.

If  \(\left( \frac { 3 }{ 2 } ,5 \right) ,\left( 7,\frac { -9 }{ 2 } \right) \)and\((\frac{13}{2},\frac{-13}{2})\) are mid-points of the sides of a triangle, then find the centroid of the triangle.

The point (3, −4) is the centre of a circle. If AB is a diameter of the circle and B is (5, −6), find the coordinates of A.

the mid-point formula to show that the mid-point of the hypotenuse of a right angled triangle is equidistant from the vertices (with suitable points).

If (x, 3), (6, y), (8, 2) and (9, 4) are the vertices of a parallelogram taken in order, then find the value of x and y.

Find the centroid of the triangle whose veritices are A(6, −1), B(8, 3) and C(10, −5).

If the centroid of a triangle is at (−2, 1) and two of its vertices are (1, −6) and (−5, 2), then find the third vertex of the triangle.

The mid-points of the sides of a triangle are (5, 1), (3, −5) and (−5, −1). Find the coordinates of the vertices of the triangle.

Find the coordinates of the point which divides the line segment joining the points (3, 5) and (8, −10) internally in the ratio 3:2.

What are the coordinates of B if point P(−2, 3) divides the line segment joining A(−3, 5) and B internally in the ratio 1 : 6?