8.03 Points and midpoints
Finding the coordinates of a midpoint
The midpoint of a line segment is a point exactly halfway between the endpoints of the segment. That is, the distance from the midpoint to both of the endpoints is the same.
Exploration
Explore this applet demonstrating the midpoint between two points in the coordinate plane. Can you generalize a pattern for finding the midpoint?
The midpoint of any two points has coordinates that are exactly halfway between the $x$x values and halfway between the $y$y values. Think of it as averaging the $x$x and $y$y values of the endpoints. We can generalize the pattern using a formula as well.
If $\overline{AB}$AB has endpoints at $A\left(x_1,y_1\right)$A(x1,y1) and $B\left(x_2,y_2\right)$B(x2,y2) in the coordinate plane, then the midpoint $M$M of $\overline{AB}$AB has the coordinates
$M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$M(x1+x22,y1+y22).
Worked example
Question 1
Evaluate: Find the midpoint between $A$A$\left(3,4\right)$(3,4) and $B$B$\left(-5,12\right)$(−5,12)
Think: We need to find the $x$x- and $y$y- coordinates that are halfway between each value.
Do: Substitute the coordinates into the midpoint formula.
| $M$M | $=$= | $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$(x1+x22,y1+y22) |
The midpoint formula |
| $=$= | $\left(\frac{3+\left(-5\right)}{2},\frac{4+12}{2}\right)$(3+(−5)2,4+122) |
$\left(x_1,y_1\right)=A\left(3,4\right)$(x1,y1)=A(3,4) and $\left(x_2,y_2\right)=B\left(-5,12\right)$(x2,y2)=B(−5,12) |
|
| $=$= | $\left(-1,8\right)$(−1,8) |
Simplify |
Reflect: Does it matter which point is labeled $\left(x_1,y_1\right)$(x1,y1) and which is labeled $\left(x_2,y_2\right)$(x2,y2)? Test your conjecture by switching the values in the formula.
Practice questions
Question 2
$M$M is the midpoint of Point $A$A $\left(-2,0\right)$(−2,0) and Point $B$B $\left(2,6\right)$(2,6).
What is the $x$x-coordinate of $M$M?
What is the $y$y-coordinate of $M$M?
Therefore, plot the Point $M$M on the axes below:
Loading Graph...Two points on a coordinate plane. Point $A$A $\left(-2,0\right)$(−2,0) and Point $B$B $\left(2,6\right)$(2,6).
Question 3
Consider the midpoint of the segment with endpoints at $A$A$\left(\frac{5}{2},-\frac{5}{2}\right)$(52,−52) and $B$B$\left(\frac{9}{2},-\frac{9}{2}\right)$(92,−92).
Find its $x$x-coordinate.
Find its $y$y-coordinate.
Therefore, write down the coordinates of the midpoint of $\overline{AB}$AB.
Finding other coordinates
Suppose we are given the midpoint of a segment and one of the endpoints of the segment. How can we reverse our steps above to find the other endpoint?
We can apply algebraic techniques to solve for unknowns.
Worked example
Question 4
Solve: Find the coordinates of $B\left(x,y\right)$B(x,y) if $M\left(2,-5\right)$M(2,−5) is the midpoint of $A\left(-4,3\right)$A(−4,3) and $B$B.
Think: What information do we know? What information are we trying to find?
Do: Substitute the information we know into the midpoint formula.
| $\left(\frac{x+\left(-4\right)}{2},\frac{y+3}{2}\right)$(x+(−4)2,y+32) | $=$= | $\left(2,-5\right)$(2,−5) |
We can write two separate equations to solve for $x$x and $y$y.
| $\frac{x+\left(-4\right)}{2}$x+(−4)2 | $=$= | $2$2 |
The midpoint formula |
| $x-4$x−4 | $=$= | $4$4 |
Multiply both sides by $2$2 |
| $x$x | $=$= | $8$8 |
Add $4$4 to both sides |
| $\frac{y+3}{2}$y+32 | $=$= | $-5$−5 |
The midpoint formula |
| $y+3$y+3 | $=$= | $-10$−10 |
Multiply both sides by $2$2 |
| $y$y | $=$= | $-13$−13 |
Subtract $3$3 from both sides |
The coordinates of B are $\left(8,-13\right)$(8,−13).
Reflect: How might we check that our answer is correct?
Practice questions
Question 5
If the midpoint of $A$A$\left(a,b\right)$(a,b) and $B$B$\left(1,4\right)$(1,4) is $M$M$\left(9,7\right)$(9,7):
Find the value of $a$a.
Find the value of $b$b.
Question 6
Find the midpoint of $A$A$\left(-7m,-3n\right)$(−7m,−3n) and $B$B$\left(-5m,-5n\right)$(−5m,−5n).
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