Transformations
For example, each point of D DEF has moved 9 units to the right and 3 units up from each point on D ABC. D DEF is the image of D ABC.
Notice that in a translation, the image is congruent to the original figure.
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Example 2: Translating a Figure
Animation In an animation, a kite will be translated 4 units to the right and 5 units down. The images of points A, B, C, and D will be points P, Q, R and S. Draw the image and give the coordinates of points P, Q, R and S.
Solution: To draw the image, think of sliding the original figure 4 units to the right and 5 units down.
You'll get the same image if you add 4 to the x–coordinates and subtract 5 from the y–coordinates.
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Answer The coordinates are P(6, 4), Q(9, 5), R(8, 2) and S(5, 1). REFLECTION
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In a reflection, the original figure is flipped over a line to produce a congruent mirror image. The line is called the line of reflection.
Example: Identifying Reflections
Tell whether the red figure is a reflection of the blue figure. If it is a reflection, identify the line of reflection.
a. |
b. |
c. |
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Yes, The line of reflection is the x–axis. |
Yes. The line of reflection is the y–axis |
No. The figure is not flipped. |
Rotations
The blue figure below was turned 90 ° clockwise about the origin to produce the congruent red image. The diagram illustrates a rotation. In a rotation, a figure is rotated through a given angle about a fixed point called the center of rotation. The angle is called the angle of rotation. In this book, all rotations are clockwise rotations about the origin.
Example: Identifying Rotations
Tell whether the red figure is a rotation of the blue figure about the origin. If it is a rotation, state the angle of rotation.
a. |
b. |
c. |
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Yes. The figure is rotated 90 ° . |
No. This is a flip, not a turn. |
Yes. The figure is rotated 180 ° . |
Tessellations
A tessellation is a repeating pattern of figures that fill a place with no gaps or overlaps.
A regular tessellation is made from only one type of regular polygon. For example, the kitchen floor tiles below suggest a regular tessellation.
Example: Forming Regular Tessellations
Tell whether the polygon can form a regular tessellation.
a. regular pentagon b. regular hexagon
Solution: a. Start with a regular pentagon. Make two copies and fit the pentagons together as shown. The gap around their common vertex cannot be filled by a fourth regular pentagon. So, regular pentagons cannot form a regular tessellation.
b. Start with a regular hexagon. Make six copies and fit the hexagons together as shown. The resulting pattern will fill a plane with no gaps or overlaps. So, regular hexagons can form a regular tessellation.
Example: Forming Tessellations
Draw a tessellation of the scalene triangle shown.
Solution: Step 1 Locate and mark a point at the middle of one side of the triangle. Rotate the triangle 180 ° about the point to form a parallelogram.
Step 2 Translate the parallelogram as shown so that the pattern fills the plane with no gaps or overlaps.
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