PROBLEMS INVOLVING MIDPOINT FORMULA WITH UNKNOWN POINTS

What is midpoint ?

The point which lies exactly in the middle of the line segment joining two points is called midpoint. 

Here A is (x₁, y₁) and B is (x₂, y₂).

Problem 1 :

If the distance between the points (4, k) and (1, 0) is 5, then what can be the possible values of k?

Solution:

Let us consider the given points as A(4, k) and B(1, 0). It is given that the distance AB is 5 units.

By distance formula,

Hence, values of k are 4, -4.

Problem 2 :

Find the value of a, for which point P(a/3, 2) is the midpoint of the line segment joining the points Q(-5, 4) and R(-1, 0).

Solution:

Midpoint of the line segment joining the points Q and R is P.

Equating the x-coordinates,

-6/2 = a/3

-3 = a/3

a = -9

Problem 3 :

In 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

A) 2      B) 4     C) -4        D) -5/2

Solution:

Q is the midpoint of P and B.

x₁ = 5, y₁ = -3

x₂ = 1, y₂ = -5

(5+1)/2, (-3+(-5))/2 = (3, y)

(6/2, -8/2) = (3, y)

(3, -4) = (3, y)

So, the value of y is -4. Option C is correct.

Problem 4 :

If P(2, p) is the mid-point of the line segment joining the points A(6, -5) and B(-2, 11), find the value of p.

Solution:

So, value of p is 3.

Problem 5 :

Find the value of k if P(4, -2) is the mid point of the line segment joining the points A(5k, 3) and B(-k, -7).

Solution:

The coordinates of the midpoint of (x₁, y₁) and (x₂, y₂) is

Equating x coordinates,

2k = 4

k = 2

Problem 6 :

If the mid-point of the line segment joining the points P(6, b-2) and Q(-2, 4) is (2, -3), find the value of b.

Solution:

The coordinates of the midpoint of (x₁, y₁) and (x₂, y₂) is

Equating the coordinates,