REVIEW

To graph ordered pairs (x,y), we need two number lines, one for each variable. The two number lines are drawn as shown below. The horizontal number line is called the x-axis and the vertical line is called the y-axis. Together, these axes form a rectangular coordinate system, also called the Cartesian coordinate system.

The coordinate system is divided into four regions, called quadrants. These quadrants are numbered counterclockwise, starting with the one in the top right quadrant. Points on the axes themselves are not in any quadrant. The point at which the x-axis and y-axis meet is called the origin. This is the point corresponding to (0, 0). The x-axis and y-axis determine a plane— a flat surface illustrated by a sheet of paper. By referring to the two axes, we can associate every point in the plane with an ordered pair. The numbers in the ordered pair are called the coordinates of the point. In a plane, both numbers in the ordered pair are needed to locate a point. The ordered pair is a name for the point.

For example, locate the point associated with the ordered pair (2,3) by starting at the origin. Since the x-coordinate is 2, go 2 units to the right along the x-axis. Since the y-coordinate is 3, turn and go up 3 units on a line parallel to the y-axis. The point (2,3) is plotted in the figure to the right. From now on the point with x-coordinate 2 and y-coordinate 3 will be referred to as point (2,3).

EXAMPLE: Plot the given points in a coordinate system: Plotting Ordered Pairs

TRANSFORMATIONS

Learn to transform plane figures using: Translations Rotations Reflections.

transformation translation rotation center of rotation reflection image Vocabulary

When you are on an amusement park ride, you are undergoing a transformation. Ferris wheels and merry-go-rounds are rotations. Free fall rides and water slides are translations. Translations, rotations, and reflections are type of transformations.

The resulting figure or image, of a translation, rotation or reflection is congruent to the original figure.

Identifying Transformations Identify each as a translation, rotation, reflection, or none of these. A. B. reflection rotation

Identifying Transformations Identify each as a translation, rotation, reflection, or none of these. C. D. none of the thesetranslation

Identify each as a translation, rotation, reflection, or none of these. A B C A. B. A B C D A’ B’ C’ D’ translation reflection A’A’ B’B’ C’C’

Identify each as a translation, rotation, reflection, or none of these. B C D E F C. D. A A’A’ B’B’ C’C’ D’D’ F’F’ E’E’ rotation none of these

Draw the image of the triangle after the transformation. A B C A. Translation along AB so that A’ coincides with B A’ B’ C’

Draw the image of the triangle after the transformation. A B C B. Reflection across BC. A’ B’ C’

Draw the image of the triangle after the transformation. A B C C. 90° clockwise rotation around point B A’ B’ C’

Draw the image of the polygon after the transformation. A B C D E F A’ B’ C’ D’ E’ F’ A. Translation along DE so that E’ coincides with D

Draw the image of the polygon after the transformation. A B C D E F B. Reflection across CD. A’ B’ C’ D’ E’ F’

Draw the image of the polygon after the transformation. A B C D E F C’ A’ B’ D’ E’F’ C. 90° counterclockwise rotation around point C

Draw the image of a triangle with vertices of (1, 1), (2, –2 ), and (5, 0) after each transformation. A. 180° counterclockwise rotation around (0, 0)

Draw the image of a triangle with vertices of (1, 1), (2, –2 ), and (5, 0) after each transformation. B. Translation 5 units left

Draw the image of a triangle with vertices of (1, 1), (2, –2 ), and (5, 0) after each transformation. C. Reflection across the x-axis

Draw the image of a shape with vertices of (1, –2), (3, 2), (7, 3), and (6, –1) after each transformation. A. 180° clockwise rotation around (0, 0) x y –2 2

B. Translation 10 units left Draw the image of a shape with vertices of (1, –2), (3, 2), (7, 3), and (6, –1) after each transformation. x y –2 2

C. Reflection across the x-axis Draw the image of a shape with vertices of (1, –2), (3, 2), (7, 3), and (6, –1) after each transformation. x y –2 2

Given the coordinates for the vertices of each pair of quadrilaterals, determine whether each pair represents a translation, rotation, reflection, or none of these. 1. (2, 2), (4, 0), (3, 5), (6, 4) and (3, –1), (5, –3), (4, 2), (7, 1) 2. (2, 3), (5, 5), (1, –2), (5, –4) and (–2, 3), (–5, 5), (–1, –2), (–5, –4) translation reflection

Given the coordinates for the vertices of each pair of quadrilaterals, determine whether each pair represents a translation, rotation, reflection, or none of these. 3. (1, 3), (–1, 2), (2, –3), (4, 0) and (1, –3), (–1, 2), (–2, 3), (–4, 0) 4. (4, 1), (1, 2), (4, 5), (1, 5) and (–4, –1), (–1, –2), (–4, –5), (–1, –5) none rotation

Dilation

Your pupils are the black areas in the center of your eyes. When you go to the eye doctor, the doctor may dilate your pupils, which makes them larger.

Translations, reflections, and rotations are transformations that do not change the size or shape of a figure. A dilation is a transformation that changes the size, but not the shape, of a figure. A dilation can enlarge or reduce a figure.

A scale factor describes how much a figure is enlarged or reduced. A scale factor can be expressed as a decimal, fraction, or percent. A 10% increase is a scale factor of 1.1, and a 10% decrease is a scale factor of 0.9.

A scale factor between 0 and 1 reduces a figure. A scale factor greater than 1 enlarges it. Helpful Hint

Tell whether each transformation is a dilation. The transformation is a dilation. The transformation is not a dilation. The figure is distorted. Additional Example 1A & 1B: Identifying Dilations A. B.

Tell whether each transformation is a dilation. A'A' B' B'C'C' A BC A. B A C A'A' B' B'C'C' The transformation is a dilation. The transformation is not a dilation. The figure is distorted. Try This: Example 1A & 1B B.

Tell whether each transformation is a dilation. The transformation is a dilation. The transformation is not a dilation. The figure is distorted. Additional Example 1C & 1D: Identifying Dilations C. D.

Tell whether each transformation is a dilation. C. The transformation is a dilation. The transformation is not a dilation. The figure is distorted. Try This: Example 1C & 1D A'A' B'B' C'C' A B C D. A'A' B'B' C'C' A B C

Every dilation has a fixed point that is the center of dilation. To find the center of dilation, draw a line that connects each pair of corresponding vertices. The lines intersect at one point. This point is the center of dilation.

Dilate the figure by a scale factor of 1.5 with P as the center of dilation. Additional Example 2: Dilating a Figure Multiply each side by 1.5.

Dilate the figure by a scale factor of 0.5 with G as the center of dilation. G FH 2 cm Multiply each side by 0.5. Try This: Example 2 G FH 2 cm F’H’ 1 cm

Additional Example 3A: Using the Origin as the Center of Dilation Dilate the figure in Example 3A on page 363 by a scale factor of 2. What are the vertices of the image? Multiply the coordinates by 2 to find the vertices of the image. A(4, 8) A’(4  2, 8  2) A’(8, 16) B(3, 2) B’(3  2, 2  2) B’(6, 4) C(5, 2) C’(5  2, 2  2) C’(10, 4) The vertices of the image are A’(8, 16), B’(6, 4), and C’(10, 4). ABC A’B’C’

Try This: Example 3A Dilate the figure by a scale factor of 2. What are the vertices of the image? B C A

Try This: Example 3A Continued A(2, 2) A’(2  2, 2  2) A’(4, 4) B(4, 2) B’(4  2, 2  2) B’(8, 4) C(2, 4) C’(2  2, 4  2) C’(4, 8) ABC A’B’C’ The vertices of the image are A’(4, 4), B’(8, 4), and C’(4, 8).

Try This: Example 3A Continued B’ C’ A’ B C A

Additional Example 3B: Using the Origin as the Center of Dilation Dilate the figure in Example 3B by a scale factor of. What are the vertices of the image? 1313 The vertices of the image are A’(1, 3), B’(3, 2), and C’(2, 1). ABC A’B’C’ A(3, 9) A’(3 , 9  ) A’(1, 3) B(9, 6) B’(9 , 6  ) B’(3, 2) C(6, 3) C’(6 , 3  ) C’(2, 1) Multiply the coordinates by to find the vertices of the image