Chapter 2 Big Ideas Study Guide
(Transformations)
Vocabulary: Use the glossary in the back of your workbook to write out the definitions for the following:
Congruent figures
Corresponding angles
Corresponding sides
Transformations
Image
Translation
Reflection
Line of Reflection
Rotation
Center of Rotation
Angle of rotation
Similar figures
Dilation
Center of dilation
Scale Factor
What you Learned
:A coordinate plane formed by the intersection of two number lines and is used to find exact locations. The plane is divided into 4 quadrants. You can graph or name any point on a coordinate plane by using an ordered pair. In an ordered pair (x,y) the x always comes before the y (remember you go in the elevator before you go up or down). The x coordinate tell how far to move horizontally (left or right) along the x-axis. The y coordinate tells how far to move vertically (up or down) along the y-axis.
Steps to graphing ORDERED PAIRS: (x,y)
1. The first # in an ordered pair tells how far to move horizontally ( , MOVE LEFT / RIGHT 1st)
2. The second # in an ordered pair tells how far to move vertically ( , MOVE UP / DOWN 2nd).
3. When naming a point on a grid, be sure to write the #’s in the correct order. Always write , (never ,).
4. Put parentheses around the #’s to indicate that it is an ordered pair.
Reflections
: a reflection is “mirror” image of the original figure. If you were to fold your paper then, the coordinates would touch. Reflections across the X-axis – keep the x-coordinate, use the opposite of the y-coordinate
Reflections across the Y-axis – keep the y-coordinate, use the opposite of the x-corrdinate
Lesson 1:
Two figures are congruent when they have exactly the same size and same shape. Matching sides are called corresponding sides. Matching angles are called corresponding angles. Two figures are congruent when corresponding sides and corresponding angles are equal.
Naming Corresponding Parts:
Corresponding angles are matching angles. Name them with a the angle symbol (<) & vertex
Corresponding sides are matching sides. Name them with two letters.
Corresponding Angles: Corresponding Sides:
<D = <A DE = AC
<F = <B EF = CB
Identifying Congruent Figures:
Corresponding sides and corresponding angles must be equal for figures to be congruent.
Angles are equal, but sides are not. These are not congruent.
Using Congruent Figures
If two figures are congruent corresponding sides will be equal, This means that the area and perimeter will equal.
Lesson 2:
A transformation changes a figure into another figure. The new figure is called the image. The original figure is named A, B, or C while the image is names A’, B’, or C’ (read as A prime, B prime, or C prime). In a translation you slide (move) a figure from one place to another without turning it. Every point of the figure moves the same distance & in the same direction. Slides (translations) can move in a horizontal, vertical, or diagonal direction, but the size/shape/direction of the figures do not change (so the figures remain congruent).Types of Transformations
Rotation Turn!
Reflection Flip!
Translation Slide!
After any of those transformations (turn, flip or slide),the shape is congruent because it still has the same size, area, angles and line lengths.
Resizing/Similar Shapes
The other important Transformation is Resizing (also calleddilation, contraction, compression, enlargement or
even expansion). The shape becomes bigger or smaller which is means it is similar but no longer
Identifying a Translation –
In a translation the original figure will slide from one place to another but will not turn or flip. The size, shape, and direction of the image will be the same.
Translation Translation Not a translation (figure turned)
Translations in a Coordinate Plane:
Method 1 (Using graph):1. Graph the original image in the coordinate plane 2. Graph the translation
3. Name the new set of coordinates for the image Method 2 (Using Coordinates):
1. Write the coordinates of the original image. Remember coordinates are written in the form (x,y).
2. Add the horizontal # to the x coordinate. Add a positive to move right & a negative to move left. 3. Add the vertical # to the y coordinate. Add a positive to move up & a negative to move down. 4. Use the new coordinates to graph the image.
Method 1 - Move each coordinate 7 units left/3 down
Method 2 – Add the movement to the coordinates: x + (-7) and y + (-3)
A = (2,4) A’ = (2 + (-7) , (4 + (-3) = (-5,1)
B = (4,4) B’ = (4 + (-7), (4 + (-3) = (-3,1) C = (5,2) C’ = (5 + (-7), (2 + (-3) = (-2, -1) D = (2,1) D’ = (2+(-7), (1 + (-3) = (-5,-2)
Describing a Translation:
1. Tell how many units the figure moved horizontally (side/side).
2. Tell how man units the figure moved vertically (up/down).
Lesson 3:
A reflection is a mirror image (or flip). If you were to fold a paper across the line of reflection, then the coordinates would touch. Reflections can be made across the x-axis, the y-axis, or any other coordinate (which would be a line of reflection).Reflection across line Reflection across the x & y-axis
Identifying a Reflection
If you fold the paper across the line of reflection, would the images match/line up. If the answer is yes, then they are a reflection. If the answer is no, then they are not reflections.
Yes / Reflection Not a reflection (this is a translation/slide instead)
Reflection A Figure Across the Axis:
Reflections across the X-axis – keep the x-coordinate, use the opposite of the y-coordinate (x,y) (x, -y)
Reflections across the Y-axis – keep the y-coordinate, use the opposite of the x-corrdinate (x,y) (-x, y)
Reflect across the x axis (keep x, opposite y) Reflect across the y axis (keep y, opposite x)
Lesson 4:
A rotation is when the figure turns about a point. The point in which the figure turns is called the center of rotation which may or may not be attached to the figure. A figure can turn clockwise (to the right) or counterclockwise (to the left). The number of degrees a figure turns is the angle of rotation. In a rotation, the original figure and the image are congruent.Rotating A Figure:
Think of how the image would be positioned if you were to cut it out then turn it. 1. Decide what the point of rotation is.2. Decide which direction the object has rotated (clockwise or counterclockwise). 3. Decide how many degrees to rotate/turn.
A = original image
D = 270° clockwise rotation
Using More Than One Transformation:
Complete each transformation one step at a time.Reflect the figure across the y axis. Then transform it 3 units up.
Rotation Rules
Lesson 5:
Figures that have the same shape, but not the same size are called similar. In similar figures the image is either larger or smaller than the original (in other words it is resized). Two figures are similar when the corresponding sides are proportional and the corresponding angles are equal. The symbol means similar. The symbol means congruent.Triangle ABC Triangle DEF
<A = <D AB = BC = AC
<B = <E DE EF DF (see below) <C = <F
Checking To See If Sides Are Proportional
: In a proportion, two ratios are equal. 1. Write the corresponding sides of each figure as a ratio. (original figure / image) 2. Simplify each ratio.3. If ratios are equal then the sides are proportional. If ratios are not equal then sides are not proportional. Using example above:
AB = 10 = 1 BC = 6 = 1 AC = 7 = 1 *All of these simplify to ½ so sides DE 20 2 EF 12 2 DF 14 2 are proportional
Identifying Similar Figures:
See Example Above – Angles are equal. Sides are proportional
Finding An Unknown Measure:
1. Since corresponding sides are proportional, use the two corresponding sides to write a proportion. Be sure to set up the proportion so that the corresponding sides are compared (original/image).
2. Use a variable for the unknown side.
3. Cross multiply. Multiply the numerator of one side by the denominator of the other. 4. Simplify
left vs.right
original 8 = 6 18 x 6 = 8x
image 18 x 108 = 8x
x = 13.5 in
Lesson 6:
When two figures are similar, the ratio of their perimeters is equal to the ratio of their corresponding sides. Also the ratio of their areas is equal to the square of the ratio of their corresponding sides.Finding the Ratio of Perimeters:
1. List the corresponding lengths of the figures (original / image)2. Simplify Example Above:
AB = BC = AC 6 = 8 = 10 *All of these simplify to 2 so the ratio of perimeters is 2
DE EF DF 3 4 5 1 1
Finding the Ratio of Areas:
1. Square the corresponding lengths of the figures. 2. Simplify
Example above:
= so the ratio of areas is 4/1
Using Proportions to Find Perimeters and Areas
1. Since corresponding sides are proportional, use the two corresponding sides to write a proportion. Be sure to set up the proportion so that the corresponding sides are compared (original/image).
2. Use a variable for the unknown.
3. Cross multiply. Multiply the numerator of one side by the denominator of the other. 4. Simplify
Perimeter Example: (perimeter / sides)
x = 15 cm
Area Example:
big (area / sides) small
16x = 49x16
16x = 784 x = 49 ft
Lesson 7:
A dilation is when an object is resized to become smaller or larger with respect to a point. When a figure is dilated lines connecting corresponding vertices will meet at a point called the center of dilation. In a dilation, the original figure and its image are similar. The ratio of the corresponding side lengths is called a scalefactor (k). To dilate a figure, multiply the coordinates by the scale factor (k). When the scale factor is greater than one the figure is an enlargement. When the scale factor is between 0-1 the dilation is a reduction.Identifying Dilation:
1. Connect the vertices of the image with the original figure using a straight line. Extend this line past the original figure.
2. If the lines meet at one precise point the image is a dilation. If they do not meet at a precise point, the image is not a dilation.
Lines connecting vertices meet at a point, so the figure is a dilation.
Not a dilation. Do not meet at one point.
Scale Factors:
1. Find the side length of the image and the original figure. 2. Write these lengths as a ratio (image/original)
3. When the scale factor is greater than one the figure is an enlargement. When the scale factor is between 0-1 the dilation is a reduction.
Scale Factor for A to B = 8 = 2 *scale factor is between 0-1 so A is smaller than B 12 3
Scale Factor for B to C = 12 = 3 *scale factor is >1, so B is larger than C
4 1
Dilating a Figure:
2. Multiply the x and y coordinates by the scale factor. 3. Use these new coordinates to graph the dilation.
Draw the image of triangle abc after a dilation with a scale factor of 2. Identify the type of dilation.
Original Mutiply by Scale Factor Vertices of Image a’b’c’ a = (1,2) (1x2 , 2x2) a’ = (2,4)
