Precalculus by Richard Wright

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# 1-01 The Cartesian Plane

Summary: In this section, you will:

• Plot points in the Cartesian coordinate system.
• Find the distance between two points.
• Find the midpoint between two points.

SDA NAD Content Standards (2018): PC.5

## Plot Ordered Pairs in the Cartesian Coordinate System

There is a legend that claims the Cartesian coordinate system was discovered when René Descartes was sick in bed. As he was staring up at the ceiling, he saw a fly. He wondered how he could describe the fly's position on the ceiling. If one corner of the room was taken as a reference point or origin, the fly's position could be described as the number of tiles horizontally from the origin followed by the number of tiles vertically from the origin. These two numbers are the coordinates of the fly's position. René Descartes first published the coordinate system that bears his name in 1637. It is sometimes called the rectangular coordinate system because the grid is made of squares.

The Cartesian, or rectangular, coordinate system consists of a horizontal x-axis and a vertical y-axis. The point where the axes cross is called the origin. Any point can be described as the distance it is from the origin along the x-axis and along the y-axis and is written as (x, y). Positive x is to the right and positive y is up. Negative x is to the left and negative y is down. The location of a point is called its coordinate.

The horizontal x-axis and the vertical y-axis divide the plane into four quadrants which are numbered counterclockwise from the top right.

###### Plot a Point in the Cartesian Coordinate System

Given the point (x, y)

1. Starting at the origin, move right the distance of x. Move left if it is negative.
2. Move up the distance of y. Move down if it is negative.

#### Example 1: Plot Points

Plot the points A(3, −2), B(1, 4), C(−3, −1), D(0, 2).

###### Solution

The first number is the x and indicates how far right to move from the origin. The second number is the y and indicates how far up to move from the origin.

• Point A(3, −2): From the origin, move 3 right and 2 down.
• Point B(1, 4): From the origin, move 1 right and 4 up.
• Point C(−3, −1): From the origin, move 3 left and 1 down.
• Point D(0, 2): From the origin, move 0 left or right and move 2 up.

Figure 4 illustrates how to graph the points. The final graph would not have the arrows; it would only have the points.

##### Try It 1

Plot the points M(3, −4) and N(−3, 0).

## Distance Formula

Now that points can located on the Cartesian plane, it might be nice to know how far apart they are. A right triangle can be created by drawing a segment between the two points. Then draw a the horizontal and vertical distances. The horizontal distance is found be subtracting the x-values of the points, Δx = x2x1, and the vertical distance is found by subtracting the y-values of the points, Δy = y2y1. The distance between the two points is the length of the hypotenuse of the right triangle and can be found using the Pythagorean Theorem.

$$d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$$

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

###### Distance Formula

The distance, d, between two points (x1, y1) and (x2, y2) is

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

#### Example 2: Use the Distance Formula

Find the distance between W(−5, 8) and E(1, −4).

###### Solution

Fill in the distance formula using

W(−5, 8) = (x1, y1) and E(1, −4) = (x2, y2)

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

$$d = \sqrt{(1 - (-5))^2 + (-4 - 8)^2}$$

$$d = \sqrt{180}$$

$$d = \sqrt{(36)(5)}$$

$$d = 6\sqrt{5}$$

##### Try It 2

Find the distance between H(3, 5) and I(−1, 3).

$$2\sqrt{5}$$

#### Example 3: Find an Endpoint from Distance

Find the missing coordinate given the distance between the points (2, 4) and (−1, y) is $$3\sqrt{2}$$.

###### Solution

Start by filling in the points and distance into the distance formula.

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

$$3\sqrt{2} = \sqrt{(-1 - 2)^2 + (y - 4)^2}$$

Simplify the first group.

$$3\sqrt{2} = \sqrt{\color{red}{9} + (y - 4)^2}$$

Square both sides.

$$\color{red}{18} = 9 + (y - 4)^2$$

Finish solving for y.

$$9 = (y - 4)^2$$

Remember to use a ± when you take a square root while solving an equation.

$$±3 = y - 4$$

$$4 ± 3 = y$$

$$y = 1, 7$$

##### Try It 3

Find the missing coordinate given the distance between the points (−1, 6) and (x, 7) is $$\sqrt{17}$$.

x = −5, 3

## Midpoint Formula

A convenient result of the way the Cartesian coordinate system works, is that the x and y directions are completely independent from each other. Motion in the x does not affect motion in the y. This allows to calculations to be done simply by just thinking about the x and y separately.

A potentially useful thing to find on the Cartesian plane is to find the point halfway between two other points. This is called the midpoint. Since the midpoint is the middle, it can be found using the arithmetic mean or average. Because of the independence of the x and y, the average can be taken in both directions separately.

$$Midpoint = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

###### Midpoint Formula

The midpoint between two points (x1, y1) and (x2, y2) is

$$Midpoint = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

#### Example 4: Midpoint Formula

Find the midpoint between R(2, 4) and W(−3, 1).

###### Solution

Fill in the midpoint formula using

R(2, 4) = (x1, y1) and W(−3, 1) = (x2, y2)

$$Midpoint = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

$$Midpoint = \left(\frac{2 + (-3)}{2}, \frac{4 + 1}{2}\right)$$

$$Midpoint = \left(-\frac{1}{2}, \frac{5}{2}\right)$$

##### Try It 4

Find the midpoint between H(3, 5) and I(−1, 3).

(1, 4)

#### Example 5: Endpoint from Midpoint

Find the missing endpoint given one endpoint is (2, −1) and the midpoint is (5, 6).

###### Solution

Fill in the given endpoint for (x1, y1) and the midpoint into the midpoint formula.

$$Midpoint = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

$$(5, 6) = \left(\frac{2 + x_2}{2}, \frac{-1 + y_2}{2}\right)$$

Because the x and y are independent, they can be solved separately. Make an equation out of just the x-coordinates and solve it.

$$5 = \frac{2 + x_2}{2}$$

$$10 = 2 + x_2$$

$$x_2 = 8$$

Repeat for the y-coordinate.

$$6 = \frac{-1 + y_2}{2}$$

$$12 = -1 + y_2$$

$$y_2 = 13$$

The answer is (x2, y2) which is (8, 13).

##### Try It 5

Find the missing endpoint given one endpoint is (−3, 0) and the midpoint is (1, −2).

(5, −4)

##### Lesson Summary

###### Plot a Point in the Cartesian Coordinate System

Given the point (x, y)

1. Starting at the origin, move right the distance of x. Move left if it is negative.
2. Move up the distance of y. Move down if it is negative.

###### Distance Formula

The distance, d, between two points (x1, y1) and (x2, y2) is

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

###### Midpoint Formula

The midpoint between two points (x1, y1) and (x2, y2) is

$$Midpoint = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

## Practice Exercises

1. (a) If a point lies on the x-axis, what is its y-coordinate? (b) If a point lies on the y-axis, what is its x-coordinate?
2. Which quadrant contains only negative coordinates?
3. Plot the given points.

4. A(3, 0), B(−2, −4), C(−1, 3)
5. D$$\left(\frac{1}{2}, \frac{3}{2}\right)$$, E$$\left(4, -\frac{1}{2}\right)$$, F(0, −2)
6. Find the coordinates of the points in the graph.
7. Find the exact distance between the two points. Use radical form.

8. A(3, 0), B(−9, −5)
9. M(1, −2), N(2, −5)
10. C(4, −2), D(−1, 3)
11. Find the missing coordinate given the distance between the points.

12. E(2, 4), F(x, 7); d = 5
13. G(−2, 5), H(2, y); d = $$2\sqrt{5}$$
14. I(3, 0), J(x, 4); d = $$4\sqrt{2}$$
15. Find the midpoint between the two points.

16. A(3, 0), B(−9, −5)
17. M(1, −2), N(2, −5)
18. C(4, −2), D(−1, 3)
19. Find the missing endpoint given one endpoint and the midpoint.

20. endpoint (2, 4), midpoint (6, 2)
21. endpoint (3, −1), midpoint (2, 0)
22. endpoint (−2, 5), midpoint $$\left(\frac{1}{2}, -\frac{5}{2}\right)$$
23. Problem Solving

24. Jeanne and Francois were doing an experiment and obtained the following data points. Graph the points and describe the pattern.
(1, 1.5), (3, 5.5), (4, 7.5), (6, 11.5), (7, 13.5)
25. A person on a boat in Lake Michigan starts to sink. If its location is (−20, 15), a Sheriff's Department boat is at (−5, 0), and a Coast Guard boat is at (−10, 33), which boat is closer to come to the rescue?
26. A manufacturer wants its warehouse halfway between Chicago and Detroit. Where should the warehouse be located if Chicago is located at 41.8781° N, 87.6298° W and Detroit is located at 42.3314° N, 83.0458° W? What major town on I-94 is that closest to? (You will probably need to put your coordinates on a map app to find out.)

1. 0; 0
2. III
3. W(1, −4), X(−3, 1), Y(4, 0), Z(1, 5)
4. 13
5. $$\sqrt{10}$$
6. $$5\sqrt{2}$$
7. x = −2, 6
8. y = 3, 7
9. x = −1, 7
10. $$\left(-3, -\frac{5}{2}\right)$$
11. $$\left(\frac{3}{2}, -\frac{7}{2}\right)$$
12. $$\left(\frac{3}{2}, \frac{1}{2}\right)$$
13. (10, 0)
14. (1, 1)
15. (3, −10)
16. ; a line
17. The Coast Guard boat
18. 42.1048° N, 85.3378° W; Battle Creek, MI