Linear Equations
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The Midpoint of an Interval

Lesson

The midpoint of a line segment is a point exactly halfway along the segment. That is, the distance from the midpoint to both of the endpoints is the same. 

Explore this applet demonstrating the midpoint between two points. 

The midpoint of any two points has coordinates that are exactly halfway between the $x$x values and halfway between the $y$y values. 

So for points A($x_1$x1, $y_1$y1) and B($x_2$x2, $y_2$y2) the midpoint will be:

M( $\frac{x_1+x_2}{2}$x1+x22 , $\frac{y_1+y_2}{2}$y1+y22 )

Think of it as averaging the $x$x and $y$y values of the end points.

Example 1

Find the midpoint between A$\left(3,4\right)$(3,4) and B$\left(-5,12\right)$(5,12)

The midpoint will be: 

M( ($\frac{x_1+x_2}{2}$x1+x22 , $\frac{y_1+y_2}{2}$y1+y22 )

=M( $\frac{3+\left(-5\right)}{2}$3+(5)2, $\frac{4+12}{2}$4+122)

=M$\left(-1,8\right)$(1,8)

Note that it does not matter which point is labelled $\left(x_1,y_1\right)$(x1,y1) and which is labelled $\left(x_2,y_2\right)$(x2,y2).

 

Working backwards to find an end point

What if we are given the midpoint of an interval, and one of the end points of the interval? How can we reverse our steps above to find the other end point?

Example 2

$\left(2,-5\right)$(2,5) is the midpoint of A$\left(-4,3\right)$(4,3) and B. Find the coordinates of B$\left(x,y\right)$(x,y).

It would be helpful to sketch what this information looks like on the number plane, so we can anticipate where B should lie.

Now to find B, consider what we would usually do to find the coordinates of the midpoint. We would find the average of the $x$x values of A and B:

$\frac{x+\left(-4\right)}{2}$x+(4)2

and the average of the $y$y values of A and B:

$\frac{y+3}{2}$y+32

In this scenario, we are given the values of the midpoint. That is:

$\frac{x+\left(-4\right)}{2}=2$x+(4)2=2, and $\frac{y+3}{2}=-5$y+32=5

So to find the coordinates of B, we simply solve each equation for its $x$x and $y$y value:

$\frac{x-4}{2}=2$x42=2                        $\frac{y+3}{2}=-5$y+32=5

$x-4=4$x4=4                        $y+3=-10$y+3=10

$x=8$x=8                              $y=-13$y=13

B has coordinates $\left(8,-13\right)$(8,13).

 

Example 3

Point $M$M$\left(-5,-3\right)$(5,3) bisects the interval joining $A$A$\left(a,b\right)$(a,b) and $B$B$\left(-7,-9\right)$(7,9).

  1. Find the value of $a$a.

  2. Find the value of $b$b.

 

 

 

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