Name: Applications of Section Formula
Uploaded: 2019-07-19
Duration: 14 min 56 s
Description: Thorough with the section formula in three-dimensional geometry? This video will show you how the section formula can be applied in various situations in geo...

Question

The line segment joining

A (6, 3)

and

B (- 1, - 4)

is doubled in length by adding half of

A B

to each end. Find the coordinates of the new end points.

Accepted Answer

Let, $C (x_{3}, y_{3})$ and $D (x_{4}, y_{4})$ be at the equal distance from the endpoints $A$ and $B$ respectively.

Given, the line segment $A B$ is doubled in length by adding half of $A B$ to each end.

We need to find the mid-point of $A B$ .

Let the mid-point be $M$ .

The formula for the mid-point of a line segment is,

$(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$

So, the midpoint of the line segment $A B$ ,

$= (\frac{6 + (- 1)}{2}, \frac{3 + (- 4)}{2})$

$= M (\frac{5}{2}, \frac{- 1}{2})$

We know that $A$ is at the mid-point of $C M$ .

So, using mid-point formula,

$(\frac{\frac{5}{2} + x_{3}}{2}, \frac{\frac{- 1}{2} + y_{3}}{2}) = (6, 3)$

$\Rightarrow (\frac{\frac{5}{2} + x_{3}}{2}) = 6$

$\Rightarrow \frac{5}{2} + x_{3} = 12$

$\Rightarrow x_{3} = 12 - \frac{5}{2}$

$\Rightarrow x_{3} = \frac{19}{2}$

Similarly,

$\Rightarrow (\frac{\frac{- 1}{2} + y_{3}}{2}) = 3$

$\Rightarrow \frac{- 1}{2} + y_{3} = 6$

$\Rightarrow y_{3} = 6 + \frac{1}{2}$

$\Rightarrow y_{3} = \frac{13}{2}$

Hence, the co-ordinates are $C (\frac{19}{2}, \frac{13}{2})$ .

We know that $B$ is the mid-point of $M D$ .

So, using mid-point formula,

$(\frac{\frac{5}{2} + x_{4}}{2}, \frac{\frac{- 1}{2} + y_{4}}{2}) = (- 1, - 4)$

$\Rightarrow \frac{\frac{5}{2} + x_{4}}{2} = - 1$

$\Rightarrow \frac{5}{2} + x_{4} = - 2$

$\Rightarrow x_{4} = - 2 - \frac{5}{2}$

$\Rightarrow x_{4} = - \frac{9}{2}$

Similarly,

$\Rightarrow (\frac{\frac{- 1}{2} + y_{4}}{2}) = - 4$

$\Rightarrow \frac{- 1}{2} + y_{4} = - 8$

$\Rightarrow y_{4} = - 8 + \frac{1}{2}$

$\Rightarrow y_{4} = - \frac{15}{2}$

$∴ D (x_{4}, y_{4}) = (\frac{- 9}{2}, \frac{- 15}{2})$

Hence, the coordinates of the end-points are $C (\frac{19}{2}, \frac{13}{2}) and D (\frac{- 9}{2}, \frac{- 15}{2})$ .

The line segment joining $A (6, 3)$ and $B (- 1, - 4)$ is doubled in length by adding half of $A B$ to each end. Find the coordinates of the new end points.

Important Questions on Coordinate Geometry

Learn and Prepare for any exam you want !

The line segment joining A(6,3) and B(-1,-4) is doubled in length by adding half of AB to each end. Find the coordinates of the new end points.

Important Questions on Coordinate Geometry

Learn and Prepare for any exam you want !

The line segment joining $A (6, 3)$ and $B (- 1, - 4)$ is doubled in length by adding half of $A B$ to each end. Find the coordinates of the new end points.