Question

find the midpoint of the line joining (2, 3) and (- 2,1), hence show that the midpoint and the two end points are collinear​

Answer 1

Step-by-step explanation:

Given Find the midpoint of the line joining (2, 3) and (- 2,1), hence show that the midpoint and the two end points are collinear

We know that the midpoint of a pair of coordinates (x A, y A) and (x B, y B) will be  

•                    = (x A + x B / 2, y A + y B / 2)
• Now the mid point is the average of two end values.
• Therefore the midpoint of (2,3) and (-2 , 1) will be
•   (2 + (- 2) / 2 , 3 + 1 / 2)
•   = (0 , 2)
• Now A,B and C are collinear if AB + BC = AC
• So AB = √(x2 – x1)^2 + (y2 – y1)^2
•           = √(-2 – 2)^2 + (1 – 3)^2
•           = √-4^2 + (- 2)^2
•           = √16 + 4
•            = √20
• So B and C are (- 2, 1) and (0, 2)
• So BC = √(0 – (-2))^2 + (2 – 1)^2
•           = √2^2 + 1^2
•           = √5
• Now A and C are (2,3) and (0,2)
• So AC = √(0 – 2)^2 + (2 – 3)^2
•           = √(- 2)^2 + (- 1)^2
•          = √4 + 1
•         = √5
• 1. AB + BC = AC  
• So √20 + √5 = √5 is not true
• 2. AB + AC = BC
• √20 + √5 = √5 is not true
• 3. BC + AC = AB
• √5 + √5 = √20
•  2 √5 = 2√5 is true

Answer 2

SOLUTION:

The given points are A (2, 3) and B (- 2, 1)

Then the coordinates of the mid-point of the line AB joining A, B are

( (2 - 2)/2, (3 + 1)/2 )

i.e., (0, 2)

The equation of the line AB is

(y - 3)/(3 - 1) = (x - 2)/(2 + 2)

or, (y - 3)/2 = (x - 2)/4

or, 4 (y - 3) = 2 (x - 2)

or, 4y - 12 = 2x - 4

or, 2x - 4y + 8 = 0

or, x - 2y + 4 = 0 ..... (1)

In order to show the mid-point (0, 2) lying on AB, we satisfy (1) no. equation by (0, 2)

The L.H.S. of (1)

= 0 - 2 (2) + 4

= - 4 + 4

= 0 = R.H.S. of (1)

Since (0, 2) satisfies the equation (1), the mid-point and the two end points are collinear.

Thus proved.