2.1 The rectangular coordinate systems and graphs  (Page 5/21)

 Page 5 / 21

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, $\text{\hspace{0.17em}}\left({x}_{1},{y}_{1}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left({x}_{2},{y}_{2}\right),$ the midpoint formula states how to find the coordinates of the midpoint $\text{\hspace{0.17em}}M.$

$M=\left(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2}\right)$

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

Finding the midpoint of the line segment

Find the midpoint of the line segment with the endpoints $\text{\hspace{0.17em}}\left(7,-2\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(9,5\right).$

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

$\begin{array}{l}\left(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2}\right)=\left(\frac{7+9}{2},\frac{-2+5}{2}\right)\hfill \\ \phantom{\rule{6.5em}{0ex}}=\left(8,\frac{3}{2}\right)\hfill \end{array}$

Find the midpoint of the line segment with endpoints $\text{\hspace{0.17em}}\left(-2,-1\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-8,6\right).$

$\left(-5,\frac{5}{2}\right)$

Finding the center of a circle

The diameter of a circle has endpoints $\text{\hspace{0.17em}}\left(-1,-4\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,-4\right).\text{\hspace{0.17em}}$ 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.

$\begin{array}{c}\left(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2}\right)\\ \left(\frac{-1+5}{2},\frac{-4-4}{2}\right)=\left(\frac{4}{2},-\frac{8}{2}\right)=\left(2,-4\right)\end{array}$

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] .

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.

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

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.

When using the distance formula $\text{\hspace{0.17em}}d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}},$ explain the correct order of operations that are to be performed to obtain the correct answer.

Algebraic

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

$y=-3x+6$

The x- intercept is $\text{\hspace{0.17em}}\left(2,0\right)\text{\hspace{0.17em}}$ and the y -intercept is $\text{\hspace{0.17em}}\left(0,6\right).$

$4y=2x-1$

$3x-2y=6$

The x- intercept is $\text{\hspace{0.17em}}\left(2,0\right)\text{\hspace{0.17em}}$ and the y -intercept is $\text{\hspace{0.17em}}\left(0,-3\right).$

Questions & Answers

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Mark Reply
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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jai
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Propessor
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jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
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Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
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Miranda
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Jeffrey
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Miranda
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Miranda
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Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
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Miranda
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Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
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Jeffrey
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Miranda
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Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
cos(- z)=cos z
Mustafa
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
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Chinni
hello
Ranjay
hi
ANSHU
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Chinni
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Chinni
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Hassan
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SORIE
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Abdel
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SORIE
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Yaona
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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