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

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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 = (\frac{x_{1} + x_{2}}{2}, \frac{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.

\begin{array}{l} (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}) = (\frac{7 + 9}{2}, \frac{- 2 + 5}{2}) \\ = (8, \frac{3}{2}) \end{array}

Try It

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

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

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.

\begin{matrix} (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}) \\ (\frac{- 1 + 5}{2}, \frac{- 4 - 4}{2}) = (\frac{4}{2}, - \frac{8}{2}) = (2, −4) \end{matrix}

Media

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.

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 $d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}},$ explain the correct order of operations that are to be performed to obtain the correct answer.

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 $(0, 6) .$

$4 y = 2 x - 1$

$3 x - 2 y = 6$

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

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Source: OpenStax, Selected topics in algebra. OpenStax CNX. Sep 02, 2015 Download for free at http://legacy.cnx.org/content/col11877/1.2

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