Coordinate Geometry and Transformations
In this lesson you will use the Cartesian coordinate plane to graph geometric figures, find their area,
and examine their location after a transformation. The transformation may be a translation or a
After completing this lesson, you should be able to
5.1 use the Cartesian coordinate system;
5.2 use coordinate systems to solve real-life problems;
5.3 identify polygons on a coordinate plane;
5.4 find the area of a polygon drawn on a coordinate plane;
5.5 translate figures on a coordinate plane;
5.6 rotate figures on a coordinate plane; and
5.7 recognize translational and rotational symmetry.
Unit 4, Section 4-1 through 4-4
Section 4-1: Coordinates for Locations pages 183–185
The Cartesian coordinate system will be used in all of your future math courses. The horizontal axis
is the x-axis. The vertical axis is the y-axis. The left side of the x-axis has negative values and the
right side has positive values. The top side of the y-axis has positive values and the bottom side has
negative values. The two axes cross at zero. The intersection of the axes is called the origin. The
axes break the coordinate system into four quadrants: I, II, III, and IV.
A coordinate is a point located anywhere on the coordinate system. A coordinate consists of (x, y).
Examine the following coordinate system. Give the coordinates of each point. The coordinates of
points A and B is given to you. Make sure you write the coordinate down before you check it. If your
numbers are in a different order than mine, your coordinate is incorrect. This is evident by looking at
the coordinates of points A and B.
Point A(4, 1). Point B(1, 4).
C(−2, − 1); D(0, − 4); E(−1, − 4); F(0, 3); G(−2, 1); H(−4, 0); I(3, 0); J(2, − 2).
The coordinate system is used in our daily lives in maps, archeological digs, painting, crime scenes,
etc. This section of your textbook uses real-life coordinate systems. Make sure to work through the
Talk it Over and Sample problems. Enjoy your assignment.
Visit the WWW Sites page and try a game involving the coordinate axis.
Complete 1–23 odd and 29–31 odd beginning on page 185 of your textbook. You can also get more
work with the Cartesian coordinate system on page 660 of your textbook. Most of your answers can
be checked in the back of the textbook. Problems 17, 19, 21, 23, 29, and 31 can be checked here.
Section 4-2: Introduction to Coordinate Geometry pages 190–194
Coordinate Geometry places geometric figures on a Cartesian coordinate plane. You may find it
helpful to review geometric figures in section 1-7 beginning on page 44 of your textbook. After
working through the Talk it Over and Sample problems, try the following examples.
Example 1: Let's try problem 6 on page 194 of your textbook.
Plot the given points carefully. Check your graph.
The figure is an acute triangle. The measure of angle B is less than 90°.
Example 2: Try problem 12 on page 194 of your textbook.
How will you break the quadrilateral into parts? I would break it into four congruent parts: Triangles
ROQ, RON, PON, and POQ.
Find the area.
In triangle ROQ, the height is 3 and the base is 5:
Areao f 4ROQ = 1 / 2 (5)(3) = 7.5.
The entire quadrilateral has four of those triangles:
Areao f quadrilateral = (4)(7.5) = 30.
The area is 30 square units.
Example 3: Try problem 14 on page 195 of your textbook.
Do you have a plan? Hint: Use a right triangle and method used on page 193 of your textbook.
Areao f 4ADC = 1 / 2 (6)(6) = 18
Areao f 4ADB = 1 / 2 (6)(2) = 6
Areao f 4ABC = Areao f 4ADC − Areao f 4ADB = 18 − 6 = 12
Areao f 4ABC = 12
For more work with area see the WWW Sites page.
Complete the odd-numbered problems, 7–17, 20, and 22–26 all beginning on page 194 of your
textbook. Most of your answers can be checked in the back of the textbook. You can check problems
20a and 20b here.
Section 4-3: Translations pages 197–199
A translation slides an object to a different position. The object does not change size or direction.
The original object is called the preimage. The translated object is called the image.
If square ABCD is translated, its image will be named A 0 B 0 C 0 D 0 . The apostrophes show this is the
object after the translation. Symbolically, the translation of ABCD to A 0 B 0 C 0 D 0 is written,
ABCD → A 0 B 0 C 0 D 0 .
Example 1: Let's try problem 2 on page 199 of your textbook.
The problem asks you to translate 4T DE 3 units down. So every vertex will move 3 units down.
You can also decrease the y value of each coordinate by 3. The y value is changed because the
translation follows the y-axis, down.
E(0, 2 − 3) = E 0 (0, − 1)
T(−1, − 1 − 3) = T 0 (−1, − 4)
D(3, − 2 − 3) = D 0 (3, − 5)
Example 2: Try problem 4 on page 199 of your textbook.
Which values in the coordinates will change, and how?
The movement 1 unit to the right will increase the x value by 1.
The movement 2 units down will decrease the y value by 2.
Example 3: Try problem 6 on page 199 of your textbook.
It is a translation. The two triangles are the same size and they are turned the same direction. Check
your description of the translation.
3 units right and 3 units down.
Translational symmetry exists when a pattern is formed by repeating a pattern. Taking an object and
translating it repeatedly.
Example 4: Look at problem 12 on page 200 of your textbook.
This design does not have translational symmetry. The colors are a definite giveaway. The colors are
A translation can be described using symbols. When a translation moves an object 3 units to the left,
the x value of all the coordinates decreases by 3. So all ordered pairs on the object (x, y) will change
to (x − 3, y). When a translation moves an object 2 units up, all the y values of all of the coordinates
will increase by 2. So all ordered pairs on the object (x, y) will change to (x, y + 2). How would a
translation of 4 units right and 3 units down be written?
(x, y) → (x + 4, y − 3)
Right affects the x-axis. Down affects the y-axis.
Example 4: Given B(2, − 3), what will be the coordinate of B 0 after the translation
(x, y) → (x + 4, y − 6)?
(x + 4, y − 6)
Replace the x and y with values from (2, − 3):
(2 + 4, − 3 − 6).
(6, − 9)
B 0 (6, − 9).
Example 5: Given B(−1, 4), what will be the coordinate of B 0 after the translation
(x, y) → (x, y + 2)?
(x, y + 2)
Replace the x and y with values from (−1, 4):
(−1, 4 + 2).
B 0 (−1, 6).
For more work with transformations see the WWW Sites page.
Complete the odd-numbered problems, 1–21 and 25–31 all beginning on page 199 of your textbook.
Most answers can be checked in the back of the textbook. Problems 1, 11, and 13 can be checked
Section 4-4: Rotations pages 202–206
A rotation is similar to a translation in that there is a preimage and an image. The preimage does not
change its size as it is rotated. It is different because the direction of the preimage has changed. The
preimage does not just slide from one location to another. The preimage is rotated about a point that
is called the center of rotation. The center of rotation can be any point on the coordinate system. The
center or rotation could even be a point on the preimage.
A rotation is measured in degrees. A rotation is also specified by the direction of the rotation,
clockwise or counterclockwise. A protractor will be a needed tool in this lesson. Work through the
Talk it Over and Sample problems on page 203–306 of your textbook.
Translational symmetry involved the repetition of a pattern by translating a shape. Rotational
symmetry involves the rotation of an object less than 360° and obtaining the original object again.
Example 1: Try problem 4 on page 206 of your textbook.
180° counter-clockwise or clockwise.
Example 2: Try problem 8 on page 207 of your textbook.
(70, 170) coordinate contains the distsance from the radio tower and degrees rotated about
radio tower. Degrees are measured from north, clockwise.
(95, 300) coordinate contains the distance from the radio tower and degrees rotated about
radio tower. Degrees and measured from north, clockwise.
c. Both systems measure rotations about the origin. Both systems measure a linear distance from
the center of rotation.
The system locating airplanes always uses a clockwise rotation. The polar graph can use a
clockwise or counterclockwise rotation. The rotation on the airplane location system always
begins at north. The rotations on the polar graph will begin wherever the preimage is located.
Example 3: Try problem 10 on page 207 of your textbook.
Graph 13° south of west. Check graph.
13° south of west.
Use the graph to determine the clockwise rotation from north.
From north to south is 180°. From south to west is 90°.
But the rotation does not go all the way to the west:
180° + 90° − 13° = 257°.
The clockwise rotation is 257°.
Divide the degrees by 10 and round the quotient to the nearest whole number. The runway number is
257 / 10 = 25.7 rounded to nearest whole number is 26.
A 0 (0, 2)
B 0 (−3, − 1)
C 0 (1, − 2)
For fun playing with some rotations see the WWW Sites page.
Complete the odd-numbered problems, 1–15, 23 and 24 beginning on page 206 of your textbook.
Most answers can be checked in the back of the textbook. Problems 9 and 11 can be checked here.
It is now time to take your fifth progress evaluation. You will be responsible for terms and concepts
reviewed in this lesson on the progress evaluation. Maybe you would like to review the terms and
concepts with a short quiz? You can also review using Checkpoint on page 210 of your textbook
and the Review and Assessment problems 2–10 on page 235 of your textbook.
1. Given the vertices of 4ABC, A(2, 1), B(−1, − 2), C(3, − 3), graph it.
A. Perform the indicated transformations.
B. Translate (x, y) → (x − 2, y + 1).
C. Rotate 90° counter-clockwise around the origin.
2. Graph the following points on a Cartesian coordinate system:
A(2, 3), B(−3, − 2), C(−2, 3), D(3, 0).
3. Find the area of ABCD.
Answers to the quiz can be found here.