United States of AmericaPA
High School Core Standards - Geometry Assessment Anchors

# 1.03 Points and midpoints

Lesson

## 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.

Midpoint formula in the coordinate plane

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​+x2​2​,y1​+y2​2​) 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).

1. What is the $x$x-coordinate of $M$M?

2. What is the $y$y-coordinate of $M$M?

3. Therefore, plot the Point $M$M on the axes below:

##### 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).

1. Find its $x$x-coordinate.

2. Find its $y$y-coordinate.

3. 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):

1. Find the value of $a$a.

2. 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).

1. $($( $\editable{}$, $\editable{}$ $)$)

### Outcomes

#### G.2.1.2.1

Calculate the distance and/or midpoint between two points on a number line or on a coordinate plane.