Section 2.1 The Coordinate Plane

Before Starting this Section, Review

  1. 1 The number line (Section P.1 , page 6)

  2. 2 Equivalent equations (Section 1.1 , page 83)

  3. 3 Completing squares (Section 1.3 , page 109)

  4. 4 Interval notation (Section P.1 , page 8)

Objectives

  1. 1 Plot points in the Cartesian coordinate plane.

  2. 2 Find the distance between two points.

  3. 3 Find the midpoint of a line segment.

A Fly on the Ceiling

One day the French mathematician René Descartes noticed a fly buzzing around on a ceiling made of square tiles. He watched the fly and wondered how he could mathematically describe its location. Finally, he realized that he could describe the fly’s position by its distance from the walls of the room. Descartes had just discovered the coordinate plane! In fact, the coordinate plane is sometimes called the Cartesian plane in his honor. The discovery led to the development of analytic geometry, the first blending of algebra and geometry.

Although the basic idea of graphing with coordinate axes dates all the way back to Apollonius in the second century b.c., Descartes, who lived in the 1600s, gets the credit for coming up with the two-axis system we use today. In Example 2, we will see how the Cartesian plane helps visualize data on credit card interest rates.

The Coordinate Plane

  1. 1 Plot points in the Cartesian coordinate plane.

A visually powerful device for exploring relationships between numbers is the Cartesian plane. A pair of real numbers in which the order is specified is called an ordered pair of real numbers. The ordered pair (a, b) has first component a and second component b. Two ordered pairs (x, y) and (a, b) are equal, and we write (x, y)=(a, b) if and only if x=a and y=b.

Just as the real numbers are identified with points on a line, called the number line or the coordinate line, the sets of ordered pairs of real numbers are identified with points on a plane called the coordinate plane or the Cartesian plane.

We begin with two coordinate lines, one horizontal and one vertical, that intersect at their zero points. The horizontal line (with positive numbers to the right) is usually called the x-axis, and the vertical line (with positive numbers up) is usually called the y-axis. Their point of intersection is called the origin. The x-axis and y-axis are called coordinate axes, and the plane they form is sometimes called the xy-plane. The axes divide the plane into four regions called quadrants, which are numbered as shown in Figure 2.1. The points on the axes themselves do not belong to any of the quadrants.

Figure 2.1

Quadrants in a plane

Side Note

Although it is common to label the axes as x and y, other letters are also used, especially in applications. In a uv-plane or an st-plane, the first letter in the name refers to the horizontal axis; the second, to the vertical axis.

The notation P(a, b), or P=(a, b), designates the point P whose first component is a and whose second component is b. In an xy-plane, the first component, a, is called the x-coordinate of P(a, b) and the second component, b, is called the y-coordinate of P(a, b). The signs of the x- and y-coordinates for each quadrant are shown in Figure 2.1. The point corresponding to the ordered pair (a, b) is called the graph of the ordered pair (a, b). However, we frequently ignore the distinction between an ordered pair and its graph.

Example 1 Graphing Points

Graph the following points in the xy-plane:

A(3, 1), B(−2, 4), C(−3,−4), D(2,−3), and E(−3, 0).

Solution

Figure 2.2 shows a coordinate plane along with the graph of the given points. These points are located by moving left, right, up, or down starting from the origin (0, 0).

Figure 2.2

Graphing points

A(3, 1)3 units right, 1 unit upD(2,−3)2 units right, 3 units downB(−2, 4)2 units left, 4 units upE(−3, 0)3 units leftC(−3,−4)3 units left, 4 units down

Practice Problem 1

  1. Graph the following points in the xy-plane:

    P(−2, 2), Q(4, 0), R(5,−3), S(0,−3), and T(−2, 12).

Pairs of related items can be displayed as ordered pairs. For example, you might write (Jason, 12/1/1982) and (Ashlym, 4/6/1990) for the names and birthdays of two relatives. A more practical example is given next.

Example 2 Graphing Data on Credit Card Interest Rates in the U.S.

The data in Table 2.1 show the average credit card interest rates in the United States over the years 2005–2014.

Table 2.1

Year 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
Interest Rate 14.54 14.73 14.68 13.57 14.31 14.26 13.09 12.96 12.95 13.02

Graph the ordered pairs (year, interest rate), where the first coordinate represents a year and the second coordinate represents the interest rate in that year.

Solution

We let t represent the years 2005 through 2014 and % represent the interest rate in each year. Since the data start from the year 2005, we show a break in the t-axis. Alternatively, we could declare a year—say, 2004—as 0. Similar comments apply to the %-axis. The graph of the points (2005, 14.54), (2006, 14.73), …, (2014, 13.02) is shown in Figure 2.3. The figure depicts the credit card interest rate for every year since 2005.

Figure 2.3

Credit card interest rates

Practice Problem 2

  1. The data in Table 2.2 show the average mortgage interest rates for a 30-year conventional loan in the United States over the years 2005–2014.

    Table 2.2

    Year 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
    Interest Rate 5.87 6.41 6.34 6.03 5.04 4.69 4.45 3.66 3.98 4.22

    Graph the ordered pairs (year, interest rate), where the first coordinate represents a year and the second coordinate represents the interest rate in that year.

The display in Figure 2.3 is called the scatter diagram of the data. There are numerous other ways of visualizing the data. Two such ways are shown in Figure 2.4(a) and Figure 2.4(b).

Figure 2.4

Two methods of visualizing data

Scales on a Graphing Utility

When drawing a graph, you can use different scales for the x- and y-axes. Similarly, the scale can be set separately for each coordinate axis on a graphing utility. Once scales are set, you get a viewing rectangle, where your graphs are displayed. For example, in Figure 2.5, the scale for the x-axis (the distance between each tick mark) is 1, whereas the scale for the y-axis is 2. Although newer calculators, like the TI-84 plus, have two display modes (CLASSIC and MATHPRINT) both display graphs identically. Read more about viewing rectangles in your graphing calculator manual.

Figure 2.5

Viewing rectangle

Distance Formula

  1. 2 Find the distance between two points.

If a Cartesian coordinate system has the same unit of measurement, such as inches or centimeters, on both axes, we can then calculate the distance between any two points in the coordinate plane in the given unit.

Recall that the Pythagorean Theorem states that in a right triangle with hypotenuse of length c and the other two sides of lengths a and b,

a2+b2=c2,Pythagorean Theorem

as shown in Figure 2.6.

Figure 2.6

Pythagorean theorem

Suppose we want to compute the distance d(P, Q) between the two points P(x1, y1) and Q(x2, y2). We draw a horizontal line through the point Q and a vertical line through the point P to form the right triangle PQS, as shown in Figure 2.7.

Figure 2.7

Visualizing the distance formula

The length of the horizontal side of the triangle is |x2−x1|, and the length of the vertical side is |y2−y1|. By the Pythagorean Theorem, we have

[d(P, Q)]2=|x2−x1|2+|y2−y1|2d(P, Q)=|x2−x1|2+|y2−y1|2Take the square root of both sides.d(P, Q)=(x2−x1)2+(y2−y1)2|a−b|2=(a−b)2

The distance between two points in a coordinate plane is the square root of the sum of the square of the difference between their x-coordinates and the square of the difference between their y-coordinates.

Example 3 Finding the Distance Between Two Points

Find the distance between the points P(−2, 5) and Q(3,−4).

Solution

Let (x1, y1)=(−2, 5) and (x2, y2)=(3,−4). Then

x1=−2, y1=5, x2=3, and y2=−4.d(P, Q)=(x2−x1)2+(y2−y1)2Distance formula=[3−(−2)]2+(−4−5)2Substitute the values for x1, x2, y1, y2.=52+(−9)2Simplify.=25+81=106≈10.3Simplify; use a calculator.

Practice Problem 3

  1. Find the distance between the points (−5, 2) and (−4, 1).

Side Note

Remember that in general a2+b2≠a+b

In the next example, we use the distance formula and the converse of the Pythagorean Theorem to show that the given triangle is a right triangle.

Example 4 Identifying a Right Triangle

Let A(4, 3), B(1, 4), and C(−2,−5) be three points in the plane.

  1. Sketch triangle ABC.

  2. Find the length of each side of the triangle.

  3. Show that ABC is a right triangle.

Solution

  1. A sketch of the triangle formed by the three points A, B, and C is shown in Figure 2.8.

    Figure 2.8

  2. Using the distance formula, we have

    d(A, B)=(4−1)2+(3−4)2=9+1=10,d(B, C)=[1−(−2)]2+[4−(−5)]2=9+81=90=310d(A, C)=[4−(−2)]2+[3−(−5)]2=36+64=100=10.

  3. We check whether the relationship a2+b2=c2 holds in this triangle, where a, b, and c denote the lengths of its sides. The longest side, AC, has length 10 units.

    [d(A, B)]2+[d(B, C)]2=10+90Replace [d(A, B)]2 with 10and [d(B, C)]2 with 90.=(10)2=[d(A, C)]2

    It follows from the converse of the Pythagorean Theorem that the triangle ABC is a right triangle.

Practice Problem 4

  1. Is the triangle with vertices (6, 2), (−2, 0), and (1, 5) an isosceles right triangle—that is, a right triangle with two sides of equal length?

Example 5 Applying the Distance Formula to Baseball

The baseball diamond is in fact a square with a distance of 90 feet between each of the consecutive bases. Use an appropriate coordinate system to calculate the distance the ball travels when the third baseman throws it from third base to first base.

Solution

We can conveniently choose home plate as the origin and place the x-axis along the line from home plate to first base and the y-axis along the line from home plate to third base, as shown in Figure 2.9. The coordinates of home plate (O), first base (A), second base (C), and third base (B) are shown in the figure.

Figure 2.9

We are asked to find the distance between the points A(90, 0) and B(0, 90).

d(A, B)=(90−0)2+(0−90)2Distance formula=(90)2+(−90)2=2(90)2Simplify.=902≈127.28 feetSimplify; use a calculator.

Practice Problem 5

  1. Young players might play baseball in a square “diamond” with a distance of 60 feet between consecutive bases. Repeat Example 5 for a diamond with these dimensions.

Midpoint Formula

  1. 3 Find the midpoint of a line segment.

Recall that a point M on the line segment PQ¯ is its midpoint if d(P, M)=d(M, Q). See Figure 2.10. We provide the midpoint formula in the xy-plane.

Figure 2.10

Midpoint of a segment

The x-coordinate of the midpoint of a line segment is the average of the x-coordinates of the segment’s endpoints, and the y-coordinate is the average of the y-coordinates of the segment’s endpoints.

We ask you to prove the midpoint formula in Exercise 72.

Example 6 Finding the Midpoint of a Line Segment

Find the midpoint of the line segment joining the points P(−3, 6) and Q(1, 4).

Solution

Let (x1, y1)=(−3, 6) and (x2, y2)=(1, 4). Then

x1=−3, y1=6, x2=1, and y2=4.Midpoint=(x1+x22, y1+y22)Midpoint formula=(−3+12, 6+42)Substitute values for x1, x2, y1, y2.=(−1,5)Simplify.

The midpoint of the line segment joining the points P(−3, 6) and Q(1, 4) is M(−1, 5).

Practice Problem 6

  1. Find the midpoint of the line segment whose endpoints are (5,−2) and (6,−1).

Section 2.1 Exercises

Concepts and Vocabulary

  1. A point with a negative first coordinate and a positive second coordinate lies in the                    quadrant.

  2. Any point on the x-axis has second coordinate                    .

  3. The distance between the points P=(x1, y1) and Q=(x2, y2) is given by the formula d(P, Q)=                   .

  4. The coordinates of the midpoint M=(x, y) of the line segment joining P=(x1, y1) and Q=(x2, y2) are given by (x, y)=                   .

  5. True or False. For any points (x1, y1) and (x2, y2)

    (x1−x2)2+(y1−y2)2=(x2−x1)2+(y2−y1)2.

  6. True or False. The point (7,−4) is 4 units to the right and 2 units below the point (3, 2).

  7. True or False. Every point in quadrant II has a negative y-coordinate.

  8. True or False. Every point on the y-axis has zero as its x-coordinate.

Building Skills

  1. Plot and label each of the given points in a Cartesian coordinate plane and state the quadrant, if any, in which each point is located. (2, 2), (3,−1), (−1,0), (−2,−5), (0, 0), (−7,4), (0, 3), (−4,2)

    1. Write the coordinates of any five points on the x-axis. What do these points have in common?

    2. Plot the points (−2,1), (0, 1), (0.5, 1), (1, 1), and (2, 1). Describe the set of all points of the form (x, 1), where x is a real number.

    1. If the x-coordinate of a point is 0, where does that point lie?

    2. Plot the points (−1,1), (−1,1.5), (−1,2), (−1,3), and (−1, 4). Describe the set of all points of the form (−1, y), where y is a real number.

  4. What figure is formed by the set of all points in a Cartesian coordinate plane that have

    1. x-coordinate equal to −3?

    2. y-coordinate equal to 4?

  5. Let P(x, y) be a point in a coordinate plane.

    1. If the point P(x, y) lies above the x-axis, what must be true of y?

    2. If the point P(x, y) lies below the x-axis, what must be true of y?

    3. If the point P(x, y) lies to the left of the y-axis, what must be true of x?

    4. If the point P(x, y) lies to the right of the y-axis, what must be true of x?

  6. Let P(x, y) be a point in a coordinate plane. In which quadrant does P lie

    1. if x and y are both negative?

    2. if x and y are both positive?

    3. if x is positive and y is negative?

    4. if x is negative and y is positive?

In Exercises 15–24, find (a) the distance between P and Q and (b) the coordinates of the midpoint of the line segment PQ¯.

  1. P(2, 1), Q(2, 5)

  2. P(3, 5), Q(−2,5)

  3. P(−1,−5), Q(2,−3)

  4. P(−4, 1), Q(−7,−9)

  5. P(−1, 1.5), Q(3,−6.5)

  6. P(0.5, 0.5), Q(1,−1)

  7. P(2, 4), Q(2, 5)

  8. P(v−w, t), Q(v+w, t)

  9. P(t, k), Q(k, t)

  10. P(m, n), Q(−n,−m)

In Exercises 25–32, determine whether the given points are collinear. Points are collinear if they can be labeled P, Q, and R so that d(P, Q)+d(Q, R)=d(P, R).

  1. (0,0), (1,2), (−1,−2)

  2. (3,4), (0,0), (−3,−4)

  3. (4,−2), (−2,8), (1, 3)

  4. (9, 6), (0,−3), (3, 1)

  5. (−1,4), (3, 0), (11,−8)

  6. (−2,3),  (3, 1), (2,−1)

  7. (4,−4), (15,1), (1,2)

  8. (1, 7), (−7,8), (−3,7.5)

  9. Find the coordinates of the points that divide the line segment joining the points P=(−4, 0) and Q=(0, 8) into four equal parts.

  10. Repeat Exercise 33 with P=(−8, 4) and Q=(16,−12).

In Exercises 35–42, identify the triangle PQR as an isosceles (two sides of equal length), an equilateral (three sides of equal length), or a scalene triangle (three sides of different lengths).

  1. P(−5, 5), Q(−1, 4), R(−4, 1)

  2. P(3, 2), Q(6, 6), R(−1,5)

  3. P(−4,8), Q(0, 7), R(−3,5)

  4. P(6, 6), Q(−1,−1), R(−5,3)

  5. P(0,−1), Q(9,−9), R(5, 1)

  6. P(−4,4), Q(4, 5), R(0,−2)

  7. P(1,−1), Q(−1, 1), R(−3,−3)

  8. P(−0.5,−1), Q(−1.5, 1), R(3−1, 32)

  9. Show that the points P(7,−12), Q(−1,3), R(14, 11), and S(22,−4) are the vertices of a square. Find the length of the diagonals.

  10. Repeat Exercise 43 for the points P(8,−10), Q(9,−11), R(8,−12), and S(7,−11).

  11. Find x such that the point (x, 2) is 5 units from (2,−1).

  12. Find y such that the point (2, y) is 13 units from (−10,−3).

  13. Find the point on the x-axis that is equidistant from the points (−5, 2) and (2, 3).

  14. Find the point on the y-axis that is equidistant from the points (7,−4) and (8, 3).

Applying the Concepts

  1. Population. The table shows the total population of the United States in millions. (205 represents 205,000,000.) The population is rounded to the nearest million. Plot the data in a Cartesian coordinate system.

    Year Total Population (millions)
    1980 228
    1985 238
    1990 250
    1995 267
    2000 282
    2005 296
    2010 308
    2016 324

    Source: U.S. Census Bureau.

  2. Use the data from Exercise 49 and the midpoint formula for the years 2010 and 2016 to estimate the total population for 2013.

In Exercises 51–54, use the following vital statistics table.

Rate per 1000 Population
Year Births Deaths Marriages Divorces
1970 18.4 9.5 10.6 3.5
1975 14.6 8.6 10.0 4.8
1980 15.9 8.8 10.6 5.2
1985 15.8 8.8 10.1 5.0
1990 16.7 8.6 9.4 4.7
1995 14.8 8.8 8.9 4.4
2000 14.4 8.7 8.5 4.2
2005 14.2 8.1 7.6 3.6
2010 13.0 8.0 6.8 3.6
2014 12.5 7.2 6.9 3.2

Source: U.S. Census Bureau.

The table shows the rate per 1000 population of live births, deaths, marriages, and divorces from 1970 to 2014. Plot the data in a Cartesian coordinate system and connect the points with line segments.

  1. Plot (year, births).

  2. Plot (year, deaths).

  3. Plot (year, marriages).

  4. Plot (year, divorces).

In Exercises 55 and 56, assume that the graph through all of the given data points is a line segment. Use the midpoint formula to find the estimate.

  1. Spending on prescription drugs. Americans spent $326 billion on prescription drugs in 2012 and $425 million in 2016. Estimate the amount Americans spent on prescription drugs in 2014.

    (Source: IMS Institute for Healthcare Informatics)

  2. Internet users. In 2012 there were 2497 million users of the Internet worldwide. By 2016 there were 3696 million such users. Estimate the number of users of the Internet in 2014.

    (Source: http://www.internetworldstats.com)

The following graph shows the percentage of smartphone sales in the United States by two leading platforms. Use this graph for Exercises 57–60.

  1. Use the appropriate line graph to determine the percentage of Android sales in June 2013.

  2. Use the appropriate line graph to determine the percentage of iPhone sales in December 2012.

  3. For the period of June 2011 through December 2014, during which period were the Android sales at a maximum?

  4. For the period of June 2011 through December 2014, during which period were the iPhone sales at a maximum?

  5. Length of a diagonal. The application of the Pythagorean Theorem in three dimensions involves the relationship between the perpendicular edges of a rectangular block and the solid diagonal of the same block.

    In the figure, show that h2=a2+b2+c2.

  6. Distance. A pilot is flying from Dullsville to Middale to Pleasantville. With reference to an origin, Dullsville is located at (2, 4), Middale at (8, 12), and Pleasantville at (20, 3), all numbers being in 100-mile units.

    1. Locate the positions of the three cities on a Cartesian coordinate plane.

    2. Compute the distance traveled by the pilot.

    3. Compute the direct distance between Dullsville and Pleasantville.

  7. Docking distance. A rope is attached to the bow of a sailboat that is 24 feet from the dock. The rope is drawn in over a pulley 10 feet higher than the bow at the rate of 3 feet per second. Find the distance from the boat to the dock after t seconds.

Beyond the Basics

  1. Use coordinates to prove that the diagonals of a parallelogram bisect each other.

    [Hint: Choose (0, 0), (a, 0), (b, c), and (a+b, c) as the vertices of a parallelogram.]

  2. Let A(2, 3), B(5, 4), and C(3, 8) be three points in a coordinate plane. Find the coordinates of the point D such that the points A, B, C, and D form a parallelogram with

    1. AB as one of the diagonals.

    2. AC as one of the diagonals.

    3. BC as one of the diagonals.

    [Hint: Use Exercise 64.]

  3. Prove that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. [Hint: Choose (0, 0), (a, 0), (b, c), and (x, y) as the vertices of a quadrilateral and show that x=b−a, y=c.]

    1. Show that (1, 2), (−2,6), (5, 8), and (8, 4) are the vertices of a parallelogram. [Hint: Use Exercise 66.]

    2. Find (x, y) assuming that (3, 2), (6, 3), (x, y), and (6, 5) are the vertices of a parallelogram.

  5. Show that the sum of the squares of the lengths of the sides of a parallelogram is equal to the sum of the squares of the lengths of the diagonals. [Hint: Choose (0, 0), (a, 0), (b, c), and (a+b, c) as the vertices of the parallelogram.]

  6. Prove that in a right triangle, the midpoint of the hypotenuse is the same distance from each of the vertices. [Hint: Let the vertices be (0, 0), (a, 0), and (0, b).]

  7. Isosceles triangle. An isosceles triangle ABC has right angle at C and hypotenuse of length c. Find the length of each leg of the triangle.

  8. Equilateral triangles. Two equilateral triangles ABC and ABD have a common side AB of length 2a. The side AB lies on the x-axis with B=(a, 0) and midpoint at the origin O. (See the figure.) Find the coordinates of the vertices A, C, and D.

  9. Midpoint formula. Use the notation in the accompanying figure to prove the midpoint formula

    M(x, y)=(x1+x22, y1+y22).

Critical Thinking / Discussion / Writing

In Exercises 73–78, describe the set of points P(x, y) in the xy-plane that satisfies the given condition.

    1. x=0

    2. y=0

    1. xy=0

    2. xy≠0

    1. xy>0

    2. xy<0

    1. x2+y2=0

    2. x2+y2≠0

    1. x≥0

    2. y≥0

  6. Describe how to determine the quadrant in which a point lies from the signs of its coordinates.

Getting Ready for the Next Section

In Exercises 79–82, evaluate each expression for the given values of x and y.

  1. x2+y2

    1. x=12, y=12

    2. x=22, y=22

  2. (x−1)2+(y+2)2

    1. x=−1, y=1

    2. x=4, y=2

  3. x|x|+|y|y

    1. x=2, y=−3

    2. x=−4, y=3

  4. |x|x+|y|y

    1. x=−1, y=−2

    2. x=3, y=2

In Exercises 83–88, find the appropriate term that should be added so that the binomial becomes a perfect square trinomial.

  1. x2−6x

  2. x2−8x

  3. y2+3y

  4. y2+5y

  5. x2−ax

  6. x2+xy

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