<< Chapter < Page Chapter >> Page >

Using the midpoint formula

When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the midpoint formula    . Given the endpoints of a line segment, ( x 1 , y 1 ) and ( x 2 , y 2 ) , the midpoint formula states how to find the coordinates of the midpoint M .

M = ( x 1 + x 2 2 , y 1 + y 2 2 )

A graphical view of a midpoint is shown in [link] . Notice that the line segments on either side of the midpoint are congruent.

This is a line graph on an x, y coordinate plane with the x and y axes ranging from 0 to 6. The points (x sub 1, y sub 1), (x sub 2, y sub 2), and (x sub 1 plus x sub 2 all over 2, y sub 1 plus y sub 2 all over 2) are plotted.  A straight line runs through these three points. Pairs of short parallel lines bisect the two sections of the line to note that they are equivalent.

Finding the midpoint of the line segment

Find the midpoint of the line segment with the endpoints ( 7 , −2 ) and ( 9 , 5 ) .

Use the formula to find the midpoint of the line segment.

( x 1 + x 2 2 , y 1 + y 2 2 ) = ( 7 + 9 2 , 2 + 5 2 ) = ( 8 , 3 2 )
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the midpoint of the line segment with endpoints ( −2 , −1 ) and ( −8 , 6 ) .

( 5 , 5 2 )

Got questions? Get instant answers now!

Finding the center of a circle

The diameter of a circle has endpoints ( −1 , −4 ) and ( 5 , −4 ) . Find the center of the circle.

The center of a circle is the center, or midpoint, of its diameter. Thus, the midpoint formula will yield the center point.

( x 1 + x 2 2 , y 1 + y 2 2 ) ( 1 + 5 2 , 4 4 2 ) = ( 4 2 , 8 2 ) = ( 2 , −4 )
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Access these online resources for additional instruction and practice with the Cartesian coordinate system.

Key concepts

  • We can locate, or plot, points in the Cartesian coordinate system using ordered pairs, which are defined as displacement from the x- axis and displacement from the y- axis. See [link] .
  • An equation can be graphed in the plane by creating a table of values and plotting points. See [link] .
  • Using a graphing calculator or a computer program makes graphing equations faster and more accurate. Equations usually have to be entered in the form y= _____. See [link] .
  • Finding the x- and y- intercepts can define the graph of a line. These are the points where the graph crosses the axes. See [link] .
  • The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment. See [link] and [link] .
  • The midpoint formula provides a method of finding the coordinates of the midpoint dividing the sum of the x -coordinates and the sum of the y -coordinates of the endpoints by 2. See [link] and [link] .

Section exercises

Verbal

Is it possible for a point plotted in the Cartesian coordinate system to not lie in one of the four quadrants? Explain.

Answers may vary. Yes. It is possible for a point to be on the x -axis or on the y -axis and therefore is considered to NOT be in one of the quadrants.

Got questions? Get instant answers now!

Describe the process for finding the x- intercept and the y -intercept of a graph algebraically.

Got questions? Get instant answers now!

Describe in your own words what the y -intercept of a graph is.

The y -intercept is the point where the graph crosses the y -axis.

Got questions? Get instant answers now!

When using the distance formula d = ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 , explain the correct order of operations that are to be performed to obtain the correct answer.

Got questions? Get instant answers now!

Algebraic

For each of the following exercises, find the x -intercept and the y -intercept without graphing. Write the coordinates of each intercept.

y = −3 x + 6

The x- intercept is ( 2 , 0 ) and the y -intercept is ( 0 , 6 ) .

Got questions? Get instant answers now!

3 x 2 y = 6

The x- intercept is ( 2 , 0 ) and the y -intercept is ( 0 , −3 ) .

Got questions? Get instant answers now!

Questions & Answers

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
how do I attempted a trig number as a starter
Tumwe Reply
cos 18 ____ sin 72 evaluate
Het Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask