Lesson 20

 The Cartesian Coordinate System and the Distance Formula

Renee Descartes was a philosopher of the 17th century, after whom the Cartesian Coordinate System is named.  Among many other things, he is famous for starting his philosophical reasoning by noting “I think, therefore I am.”  The Cartesian Coordinate system is very simple.  If you like, read about him at Britannica.com.

Throughout this whole course, we have been working with “plane geometry.”  That is, everything we imagine lies within a plane.  Although we represent the plane and the figures it contains on a piece of paper or on a computer screen, the actual figures are not limited to that size at all.  They could be much greater or much smaller than we can represent.

Generally, we have noted the relationships among nearby shapes, without stating absolutely where they are.  The Cartesian coordinate system allows us to give points, lines and other figures exact locations.  Once we have those locations, we have an entire new language for understanding geometry.  We can prove things through the logic of geometric theorems, or we can prove things by algebraically working with the equations describing the locations.

To understand the Cartesian Coordinate system, you must recall the number line.

But a number line doesn't help us name every point on the plane.  For instance, in the diagram, points A, B and C are both very close to 4, but B is a little above it, while C is a little below.  Well have to have more information to identify points on the plane.

In the Cartesian Coordinate system, you can name every point on the plane using two number lines.  The two number lines are called axes (pronounced axe-ease, the singular is axis, and is pronounced with the final s sounding like the one in “hiss” rather than like the one in “his”).  Usually, the axis that goes from left to right across the page is called the "x-axis," and the one that goes from the bottom to the top of the page is known as the "y-axis."

Recall how we construct a circle:  we choose a center point, and we choose a radius, and, using something like a compass, we draw all the points that are distant from the center point P by the distance r.

Using the cartesian coordinate system, we can write an equation that describes where a circle lies.  All we need is a formula for distance.

Consider the point A, which has the coordinates (3,4).  If we draw the line segment from the origin to point A, and another line through A that goes to the x axis, it is clear that we have drawn a right triangle, with the line from the origin to A as the hypoteneuse.  We therefore know that the distance from the origin to point A is sqrt(3^2 + 4^2)=5.

In general, the equation that describes the points in a circle with its center at the origin is

where x is the x coordinate, y^2 is the y coordinate, and r is the radius of the circle.  For example, the equation for a circle with radius 5 with its center at the origin would be:

Verify that each of the points in this figure “satisfies” the condition given in that equation:

The Distance Formula

Then use the Pythagorean Theorem to find the length of the third side (which is the hypoteneuse of the right triangle):

c² = a² + b²

...so:

This format always holds true. Given two points, you can always plot them, draw the right triangle, and then find the length of the hypoteneuse. The length of the hypoteneuse is the distance between the two points. Since this format always works, it can be turned into a formula:

Distance Formula:

Given the two points (x₁, y₁) and (x₂, y₂), the distance between these points is given by the formula:

Don't let the subscripts scare you. They only indicate that there is a "first" point and a "second" point; that is, that you have two points. Whichever one you call "first" or "second" is up to you.

• Find the distance between the points (–2, –3) and (–4, 4).

Just plug them in to the Distance Formula:

Then the distance is sqrt(53), or about 7.28, rounded to two decimal places.

The commonest mistake made when using the Formula is to accidentally mismatch the x-values and y-values. Don't subtract an x from a y, or vice versa; make sure you've paired the numbers properly.

Also, don't get careless with the square-root symbol. If you get in the habit of omitting the square root and then "remembering" to put it back in when you check your answers in the back of the book, then you'll forget the square root on the test, and you'll miss easy points.

You also don't want to be careless with the squaring inside the Formula. Remember that you simplify inside the parentheses before you square, not after, and remember that the square in on everything inside the parentheses, including the minus sign, so the square of a negative is a positive.

In other words, if you do each step completely, instead of sloppily or in your head, then you're much more likely to get the right answers.

By the way, it is almost always better to leave the answer in "exact" form (the square root "" above). Rounding is usually reserved for the last step of word problems. If you're not sure which format is preferred, do both, like this:

Very often you will encounter the Distance Formula in veiled forms. That is, the problem will not explicitly state that you need to use the Distance Formula; instead, you have to notice that you need to find the distance, and then remember the Formula. For instance:

• Find the radius of a circle, given that the center is at (2, –3) and the point (–1, –2) lies on the circle.

The radius is the distance between the center and any point on the circle, so find the distance:

Then the radius is sqrt(10), or about 3.16, rounded to two decimal places.

The lesson section "The Distance Formula" Copyright © Elizabeth Stapel 2000-2005 All Rights Reserved

Geometry
Lesson 20
Do the test below. If you get 3 or fewer wrong, it will be assumed that you understand the lesson, and you can go on immediately to the next lesson.

If you get more than 3 wrong, you will be asked to resubmit the wrong answers and show your work. The teacher will then look at your work and give you advice on what you are doing wrong.


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Exercises

1. Draw a Cartesian Coordinate system for yourself, and then determine whether these sets of points form triangles

Set 1: (0,0), (1,1), (0,1)
Set 2: (-1, 0), (1,1), (3,2)
Set 3: (0,0), (5,0), (3,5)

A set 1 line segment, set 2 Triangle, set 3 line segment
B set 1 Triangle, set 2 Triangle, set 3 line segment
C set 1 Triangle, set 2 line segment, set 3 Triangle
D set 1 line segment, set 2 line segment, set 3 line segment
E set 1 Triangle, set 2 line segment, set 3 line segment
F set 1 line segment, set 2 Triangle, set 3 Triangle

2. I have placed a square that is 2 units long on each side so that its sides are parallel to
the x- or y-axis, but it's center is right on the origin. What are the coordinates of the corners?

A (-1,-1), (1,-1), (-1,-1), (1,-1)
B (1,1), (1,-1), (-1,1), (-1,-1)
C (-1,-1), (-1,-1), (1,-1), (1,-1)
D (1,-1), (-1,1), (-1,-1), (-1,1)
E (-1,1), (-1,1), (-1,-1), (-1,-1)
F (1,1), (1,1), (1,1), (1,1)
 

3. What shape do these four points make? (Draw from one to the next in the order given,
and then draw from the last back to the first).

(0,-1), (1,1), (0,3), (-1,1)

A A pentagon
B An octagon
C A triangle
D A square
E A rhombus
F A cube>
 

4. Now do the same with this list, and describe the shape.

(0,-1), (0,3), (1,1), (-1,1)

A A square
B Two acute angles that meet at the acute angles
C A triangle
D Two right triangles that meet at the right angles
E A rhombus
F Two obtuse triangles that meet at the hypotenuse
 

What shape is created by the following points?

5. (-1, 0), (2, 0), (2, 4), and (-1, 4)

Aa rectangle
Ba square
Ca parallelogram
Da rhombus
Ea triangle
Fa line segment
 

6. (-2, 1), (-6, 1), (-7, 6), and (-3, 6)

Aa line segment
Ba trapezoid
Ca square
Da rhombus
Ea triangle
Fa parallelogram
 

7. (1, 1), (1, -3), (9, 1), and (5, -3)

Aa line segment
Ba trapezoid
Ca square
Da rhombus
Ea triangle
Fa parallelogram
 

8. (8, 5), (2, -3), (5, -3), and (5, 5)

Aa parallelogram
Ba trapezoid
Ca square
Da rhombus
Ea triangle
Fa rectangle
 

9. (-5, 1), (2, 1), (2, -4), and (-13, -4)

Aa trapezoid
Ba line segment
Ca square
Da rhombus
Ea triangle
Fa parallelogram
 

10. (-7, 9), (-3, 13), and (-11, 13)

Aa rhombus
Ba trapezoid
Ca square
Da right triangle
Ea circle
Fa line segment
 

Find the distance between the two points:

11. (-20, -4) and (-7, -6)

A 4
B SQRT(61)
C SQRT(290)
D 5
E SQRT(173)
F SQRT(180)
 

12. (1, 1) and (-4, 1)

A 4
B SQRT(61)
C SQRT(290)
D 5
E SQRT(173)
F SQRT(180)
 

13. (-3, 22) and (-14, 35)

A 4
B SQRT(61)
C SQRT(290)
D 5
E SQRT(173)
F SQRT(180)
 

14. (9, -0) and (-3, -8)

A 4
B SQRT(61)
C SQRT(290)
D 4[SQRT(13)]
E 5[SQRT(17)]
F 3[SQRT(180)]
 

15. (1, 2) and (5, 2)

A 4[SQRT(13)]
B SQRT(61)
C SQRT(29)
D SQRT(202)
E 4
F SQRT(60)
 

What is the radius of a circle with the given center C that passes through the given point Z?

16. C (0, 0); Z (-8, 0)

A 4
B SQRT(61)
C 8
D 5
E SQRT(173)
F SQRT(180)
 

17. C (-4, -5); Z (-10, -5)

A 4
B SQRT(202)
C SQRT(290)
D SQRT(61)
E SQRT(173)
F 6
 

18. C (-5, 8); Z (-5, 4)

A 4
B SQRT(61)
C SQRT(290)
D SQRT(202)
E SQRT(173)
F 6
 

19. C (-5, 9); Z (6, 0)

A 4
B SQRT(61)
C SQRT(290)
D SQRT(202)
E SQRT(173)
F SQRT(180)
 

20. C (-7, 6); Z (-13, -6)

A 4
B SQRT(202)
C SQRT(290)
D SQRT(61)
E SQRT(173)
F SQRT(180)
 

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