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If $A = (3, - 4)$ and the midpoints of $AB, AC$ are $(2, - 1), (4, - 5)$ respectively then the midpoint of $BC$ is

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(1, 2)

(3, - 2)

(- 1, 2)

(0, - 3)

answer is B.

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Detailed Solution

It is given that

A = (3, - 4)

and the midpoints of

AB, AC

are

(2, - 1), (4, - 5)

.
We know that if a point

C (x, y)

divides the line segment

AB

joining the points

A (x_{1}, y_{1}) B (x_{2}, y_{2})

in the ratio

m : n

.
Then we have,

x = \frac{n x_{1} + m x_{2}}{m + n}

and

y = \frac{n y_{1} + m y_{2}}{m + n}

.
For midpoint of any line segment m:n = 1:1.
So, midpoint

(x, y) = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})

.
Given the midpoint of

AB = (2, - 1)

and

A = (3, - 4)

.
Let the coordinates of the point

B

(x_{2}, y_{2}) .

(\frac{3 + x_{2}}{2}, \frac{y_{2} - 4}{2}) = (2, - 1)

\Rightarrow

\frac{3 + x_{2}}{2}

2

\Rightarrow x_{2} + 3 = 4

\Rightarrow x_{2} = 4 - 3

\Rightarrow x_{2} = 1

and

\Rightarrow \frac{y_{2} - 4}{2} = - 1

{\Rightarrow y}_{2} - 4 = - 2

{\Rightarrow y}_{2} = 4 - 2

{\Rightarrow y}_{2} = 2

So,

B = (1, 2)

.
Also,

A = (3, - 4)

.
Midpoint of

= (4, - 5)

.
Let the coordinates of the point

C

(x_{3}, y_{3}) .

(\frac{3 + x_{3}}{2}, \frac{y_{3} - 4}{2}) = (4, - 5)

\Rightarrow x_{3} + 3 = 8

and

y_{3} - 4 = - 10

\Rightarrow x_{3} = 5

and

y_{3} = - 6

\Rightarrow C = (5, - 6)

Now, by using the coordinates of the point

B

and

C,

Midpoint of

BC

= (\frac{1 + 5}{2}, \frac{2 - 6}{2})

= (\frac{6}{2}, \frac{- 4}{2})

= (3, - 2)

The correct option is 2.

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