# A line segment is bisected by a line with the equation  -7 y + 3 x = 1 . If one end of the line segment is at ( 2 , 4 ), where is the other end?

May 20, 2018

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#### Explanation:

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Given :

(a) A line segment is bisected by a line.

(b) Equation of the line : $3 x - 7 y = 1$

Find :

If one end of the line segment is at $\left(2 , 4\right)$, find the other.

color(green)("Step 1 :"

Plot the point $A \left(2 , 4\right)$ on a coordinate plane.

Graph the line $3 x - 7 y = 1$ on the coordinate plane.

Construct a perpendicular line through the point $A \left(2 , 4\right)$ to the line with the equation $3 x - 7 y = 1$.

This is the shortest distance between the line and the point $A \left(2 , 4\right) .$

color(green)("Step 2 :"

Mark the point of intersection of the perpendicular line and the line with the equation $3 x - 7 y = 1$.

This is the Mid-Point(O) of the required line segment we must find, in order to locate the coordinates of the other end of the line segment.

Measure the magnitude of $\overline{A O}$.

$\overline{A O} = 3.02$ units.

Using the Mid-Point (O) as the center, construct a circle with radius being the magnitude of the part of the line segment $A O$.

Radius $= 3.02$ units.

color(green)("Step 3 :"

Extrend the line segment $A O$ with a line.

Mark the intersection of the circle and this part of the extended line.

This is our Point $B$.

Join $O B$ and measure the magnitude of $\overline{O B}$

$\overline{O B} = 3.02$ units.

Find the coordinates of the point $B$.

$B = \left(4.38 , - 1.55\right)$. This is our required answer.

Hope you find this solution process useful to your requirement.