Solving geometrical problems using midpoint formula

Question 21

Question:   P(5, 3) is the mid point of the line joining the points A(12, 9) and B(x, y). Find the value of x and y. [2 Mark]

Answer:   

A(12, 9), B(x , y), given P(5, 3) is the mid point of AB

Mid point is given by











Hence Coordinates of B are (-2, -3).

Question 22

Question:   Find the coordinates of the mid points of the sides of a triangle whose vertices are A(3, -4), B(-7, 6), C(-5, -2). [3 Mark]

Answer:   

Let P(x₁, y₁), Q(x₂, y₂), R(x₃, y₃) be the coordinates of AB, BC and CA.



By mid point formula,





= (-2, 1)

Mid point of AB, P (x₁, y₁) = (-2, 1)





= (-6, 2)

Mid point of BC, Q (x₂, y₂) = (-6, 2)



= (-1, -3)

Hence the mid points are P(-2, 1), Q(-6, 2) and R(-1, -3).

Question 23

Question:   The points A(9, 6), B(-11, 8) and C(-3, 4) are the vertices of a triangle ABC. AD is the median. Find the Co-ordinates of D. Hence find the Coordinates of the centroid. [3 Mark]

Answer:   

AD is the median The line joining the vertice A to the mid point of the of the opposite side.

Hence D is the mid point of BC.





= (-7, 6)

Centroid G is a point which is formed by the intersection of three medians. It divides the median in the ratio 2:1(from the vertex).

Hence Coordinates of G(a, b) is given by the section formula,

Let m = 2, n = 1

x₁ = 9, y₁ = 6

x₂ = -7, y₂ = 6

Question 24

Question:   The points (1, 1), (9, 3), (11, 8) and (3, 6) are the vertices of a Quadrilateral. Show that the diagonals bisect each other. Hence classify the Quadrilateral. [3 Mark]

Answer:   

Let A(1, 1), B(9, 3), C(11, 8), D(3, 6) are the vertices of the Quadrilateral.

AC and BD are the diagonals. If they bisect each other, then O(x, y) is the mid point of AC and BD. DO = OB, AO = OC.

Coordinates of O(x, y) if it is the mid point of AC







Coordinates of O(x, y), if it is the mid point of







As the values for both the mid points are same they coincide. O is the point of bisectors of diagonals. Hence diagonals bisect each other. Therefore it is a parallelogram.

Question 25

Question:   In the circle with centre O, CD is a diameter. If the Coordinates of C and D are (10, 0) and (-2, -3). Find the Coordinates of the centre O. [2 Mark]

Answer:   

CD is the diameter.

O is the centre of circle. Diameter = 2 x radius

Hence O is the mid point of the diameter.

According to Co-ordinates of mid point formula,