Using Midpoint and Distance Formulas

For any two points in a coordinate plane, the point which lies exactly half way between them can be found using the midpoint formula and the distance between the points can be calculated using the distance formula.

Concept

Midpoint

The midpoint of a line segment is the point that divides the segment into two segments of equal length.

The point

M

is the midpoint on the segment

\overline{AB}

since the distance from

A

to

M

is the same as the distance from

M

to

B.

Concept

Segment Bisector

A segment bisector is an object that passes through the midpoint of a segment, bisecting it or dividing it into two segments of equal length. The object can be a line segment, line, ray, or a plane that passes though the midpoint.

If

M

is the midpoint of

\overline{AB},

then it is a segment bisector. Also, the line

\overleftrightarrow{QR}

is a segment bisector of

AB.

Identify the segment bisector and find the length of the segment

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Exercise

Identify the segment bisector of $\overline{AB}.$ Then find the length of $\overline{AB}.$

Show Solution

Solution

To begin, notice that

\overleftrightarrow{ST}

is a segment bisector of

\overline{AB},

because it passes through the midpoint,

M.

\overline{ST}

divides

\overline{AB}

into two segments of equal length,

\overline{AM}

and

\overline{MB}.

Their lengths are given as

AB=4x-2 \quad \text{and} \quad MB=2x+1.

We can create an equation by setting the above expressions equal to each other. Then, we can solve for

x.

When we know

x,

we can find the total length of

\overline{AB}.

4x-2=2x+1

AddEqn

\text{LHS}+2=\text{RHS}+2

4x=2x+3

SubEqn

\text{LHS}-2x=\text{RHS}-2x

2x=3

DivEqn

\left.\text{LHS}\middle/2\right.=\left.\text{RHS}\middle/2\right.

x=1.5

We can now find an expression for the length of

\overline{AB}

by adding the lengths of

\overline{AM}

and

\overline{MB}.

\begin{aligned} AM = 4x-2 \Leftrightarrow AM = 4({\color{#0000FF}{1.5}})-2 \Leftrightarrow AM = 4 \\ MB = 2x+1 \Leftrightarrow MB = 2({\color{#0000FF}{1.5}})+1 \Leftrightarrow MB = 4 \end{aligned}

Finally, to find the length of

\overline{AB}

we can add

AM

and

MB.

AB=AM + MB = 4+4=8 \text{ units}

Construction

Bisect a Segment

Construction

Bisect a Segment

To bisect a segment means to draw the object that bisects it, or passes through its midpoint. Here it will be shown how to draw the line that bisects $\overline{AB}.$

Place a compass with its pointy end at one of the segment's end-points. Draw a large arc that intersects the segment further away than the midpoint.

Keep the compass set on the same length and place its pointy end at the other end of the segment. Draw another large arc across the segment.

If the arcs have been drawn large enough, they will now intersect each other twice. If they don't, extend them. Next use a ruler to draw a line through the two points where the arcs intersect each other.

The last step is to erase the arcs and draw an arrow on each end of the line.

The line bisects the line segment.

Rule

Midpoint Formula

The midpoint, $M,$ between two points, $(x_1, y_1)$ and $(x_2, y_2),$ can be determined using the midpoint formula. The coordinates for the midpoint can be seen as half of the horizontal and vertical distance.

Rule

M\left(\dfrac{x_1 + x_2}{2},\dfrac{y_1 + y_2}{2} \right)

Consider $\overline{AB}$ shown in the coordinate plane below. The horizontal and vertical distance, $\Delta x$ and $\Delta y,$ respectively, between its endpoints are shown.

The midpoint of this segment, $M,$ divides the distance between its endpoints exactly in half both horizontally and vertically. That means, the distance from each endpoint to the midpoint is half of the established distances.

Focusing first on the

x

-coordinates, the horizontal distances between the midpoint and the endpoints, are given by

\left| x_m - x_1 \right| \quad \text{and} \quad \left| x_2 - x_m \right|,

respectively. Here, the absolute values are used to ensure the distances are positive. Since these distances are equal the following relationship is true.

\left| x_m - x_1 \right| = \left| x_2 - x_m \right|

This equation can be solved to find the

x

-coordinate of the midpoint,

x_m.

|x_m - x_1| = |x_2 - x_m|

AbsPos

\left|a\right|=a

x_m - x_1 = x_2 - x_m

AddEqn

\text{LHS}+x_m=\text{RHS}+x_m

2x_m - x_1 = x_2

AddEqn

\text{LHS}+x_1=\text{RHS}+x_1

2x_m = x_2+x_1

CommutativePropAddCommutative Property of Addition

2x_m = x_1+x_2

DivEqn

\left.\text{LHS}\middle/2\right.=\left.\text{RHS}\middle/2\right.

x_m = \dfrac{x_1+x_2}{2}

Thus, the

x

-coordinate for the midpoint is

x_m = \frac{x_1+x_2}{2}.

In the same way, it can be shown that the

y

-coordinate for

M

is

y_m = \frac{y_1 + y_2}{2}.

Therefore, the midpoint of a segment is

M\left(\dfrac{x_1 + x_2}{2},\dfrac{y_1 + y_2}{2} \right).

Find the midpoint of the segment

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Exercise

The segment $\overline{AB}$ has the endpoints $A(\text{-} 1, 6)$ and $B(5, 2).$ Find its midpoint, $M.$

Show Solution

Solution

We can find the midpoint,

M

of the segment by using the midpoint formula.

M\left( \dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2} \right)

SubstitutePointsSubstitute

\left({\color{#0000FF}{\text{-} 1,6}}\right)

&

\left({\color{#009600}{5,2}}\right)

M\left( \dfrac{{\color{#0000FF}{\text{-} 1}}+{\color{#009600}{5}}}{2}, \dfrac{{\color{#0000FF}{6}}+{\color{#009600}{2}}}{2} \right)

AddTermsAdd terms

M\left(\dfrac{4}{2},\dfrac{8}{2} \right)

CalcQuotCalculate quotient

M(2,4)

The midpoint of the segment lies at

(2,4).

Method

Dividing a Segment into a Given Ratio

Sometimes, it is necessary to divide a segment into two pieces that are not of equal length. This can be done when the endpoints of the segment and the desired ratio of the lengths are known. For example, find the point $P$ that divides $\overline{AB}$ so that the ratio of $AP$ to $PB$ is $3:2.$

1

Determine the vertical and horizontal change between

A

and

B.

To begin, it is necessary to determine the vertical and horizontal change — rise and run — between the points. To do this, count the spaces between $A$ and $B$ in both directions. $\begin{aligned} \text{vertical change}(\Delta y) &: 7 \\ \text{horizontal change}(\Delta x) &: 8 \end{aligned}$

2

Determine the distance from A to P

Since the desired ratio is $3:2,$ consider dividing $\overline{AB}$ into $5$ total pieces — $\overline{AP}$ spans $3$ of those pieces, and $PB$ spans $2.$ Choosing point $A$ as a starting point, $P$ will lie $\frac{3}{5}$ of the way to $B.$

If the entire horizontal distance is $8,$ $\frac{3}{5} \cdot 8$ gives the horizontal distance from $A$ to $P$ — $4.8$ units.
If the entire vertical distance is $7,$ $\frac{3}{5} \cdot 7$ gives the vertical distance from $A$ to $P$ — $4.2$ units.

3

Plot

P

Now that the distance from $A$ to $P$ is known, $P$ can be added to the graph to so that $\overline{AB}$ is divided into a ratio of $3:2$ at $P.$

Rule

Distance Formula

Given two points in a coordinate plane, $A(x_1, y_1)$ and $B(x_2, y_2),$ the distance between them can be calculated using the distance formula.

$AB=\sqrt{\left(x_2-x_1 \right)^2+\left(y_2-y_1 \right)^2}$

This formula can be proven using the Pythagorean Theorem.

Find the distance between the points

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Exercise

Find the distance between the points $A(2,\text{-} 1)$ and $B(6 ,2).$

Show Solution

Solution

We can find the distance between two points by using the distance formula.

AB = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 }

SubstitutePointsSubstitute

\left({\color{#0000FF}{6 ,2}}\right)

&

\left({\color{#009600}{2,\text{-} 1}}\right)

AB = \sqrt{\left({\color{#0000FF}{6}}-{\color{#009600}{2}}\right)^2 + \left({\color{#0000FF}{2}}-\left({\color{#009600}{\text{-} 1}}\right)\right)^2}

SubNeg

a-(\text{-} b)=a+b

AB=\sqrt{(6-2)^2+(2+1)^2}

AddSubTermsAdd and subtract terms

AB=\sqrt{4^2+3^2}

CalcPowCalculate power

AB=\sqrt{16+9}

AddTermsAdd terms

AB=\sqrt{25}

CalcRootCalculate root

AB=5

Thus, the distance between the points is

5

units.

Method

Finding Perimeter and Area using Distance Formula

Method

Perimeter

To find the perimeter of a shape in the coordinate plane, it is necessary to see each side of the shape as a line segment. When the coordinates of the vertices are known, the length of each side can be calculated separately using the distance formula.

The length of the perimeter is then found by adding the lengths of the sides.

Method

Area

Area is usually calculated using different formulas. However, for some shapes, one formula is not enough. Consider the example shown below.

The shape can be divided into pieces so that the area of each piece can be calculated. Here, that means dividing the shape into a rectangle and a triangle.

Once the vertices are marked, the needed lengths can be found using the distance formula. Then, the area of each shape can be found using the corresponding formula. The total area is the sum of the individual areas.

Find the area and the perimeter

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Exercise

Find the area and the perimeter of the rectangle.

Show Solution

Solution

To begin, before we can calculate the area and perimeter of the rectangle, we need to know the lengths of the sides. Since the quadrilateral is a rectangle, the top and bottom have equal lengths and the sides have equal lengths. $AB = CD \quad \text{and} \quad BC = AD$ We'll find $CD,$ the length of the right side, and $AD,$ the length of the bottom. Since the coordiantes are given, we can use the distance formula. We'll determine $CD$ first.

CD = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 }

SubstitutePointsSubstitute

\left({\color{#0000FF}{5,4}}\right)

&

\left({\color{#009600}{3,6}}\right)

CD = \sqrt{\left({\color{#0000FF}{5}}-{\color{#009600}{3}}\right)^2 + \left({\color{#0000FF}{4}}-{\color{#009600}{6}}\right)^2}

SubTermsSubtract terms

CD=\sqrt{2^2+(\text{-} 2)^2}

CalcPowCalculate power

CD=\sqrt{4+4}

AddTermsAdd terms

CD=\sqrt{8}

Thus, the length of the sides of the rectangle is

\sqrt{8}

units. The length of the top and bottom can be found in the same way.

AD = \sqrt{\left({\color{#0000FF}{5}}-\left({\color{#009600}{\text{-} 1}}\right)\right)^2 + \left({\color{#0000FF}{4}}-\left({\color{#009600}{\text{-} 2}}\right)\right)^2} \Leftrightarrow AD=\sqrt{72}

Let's label the sides with their lengths.

Now that the dimensions of the rectangle are known, the area, $A$ can be found by multiplying the length and the width, and the perimeter, $P$ can be found by adding all four sides. $\begin{aligned} A & =(\sqrt{72})(\sqrt{8}) = 24 \text{ square units} \\ P & = \sqrt{72} + \sqrt{72} + \sqrt{8} + \sqrt{8} \approx 22.6 \text{ units} \end{aligned}$ Thus, the area is $24$ square units and the perimeter is approximately $22.6$ units.

Using Midpoint and Distance Formulas

Midpoint

Segment Bisector

Example

Identify the segment bisector and find the length of the segment

Construction

Bisect a Segment

Bisect a Segment

Midpoint Formula

Rule

Example

Find the midpoint of the segment

Dividing a Segment into a Given Ratio

1

2

3

Distance Formula

Example

Find the distance between the points

Finding Perimeter and Area using Distance Formula

Perimeter

Area

Example

Find the area and the perimeter