Question

A line segment AB is produced to C such that AC = 2AB. If the coordinates of A and B are (6,8) and (8,6) respectively, then the coordinates of C are

• A. (10,4)
• B. (5,2)
• C. (7,7)
• D. (8,7)

Answer 1

The correct option is A (10,4)

Given that, a line segment $AB$ is produced to $C$ such that $AC=2AB$.

Also, the coordinates are $A(6,8)$ and $B(8,6)$.

To find out: The coordinates of the point $C$.

Let the coordinates of point $C$ be $(x,y)$.

Since, $AC=2AB$, the point $C$ divides the line segment in the ratio $2:1$ externally.

We know that, if a point $P(x,y)$ divides a line segment joining $A(x_1,y_1)$ and $B(x_2,y_2)$ in the ratio $m:n$ externally, then the coordinates of the point $P$ are:

$x=\frac{m{x}_2-n{x}_1}{m-n}$ and $y=\frac{m{y}_2-n{y}_1}{m-n}$

Here, $m=2,n=1,x_1=6,y_1=8x_2=8,y_2=6$

$\therefore x=\frac{2\left(8\right)-1\left(6\right)}{2-1}$ and $y=\frac{2\left(6\right)-1\left(8\right)}{2-1}$

$\Rightarrow x=\frac{16-6}{1}$ and $y=\frac{12-8}{1}$

$\Rightarrow x=10$ and $y=4$

Hence, the coordinates of the point $C$ are $(10,4)$.