1. 13 Chapter Congruence and Similarity with Transformations
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2. 13-1 Translations and Rotations
Translations Constructions of Translations Coordinate Representation of Translations Rotations Slopes of Perpendicular Lines
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Isometries
Any rigid motion that preserves length or distance is an isometry. A translation is one type of isometry.
Any function from a plane to itself that is a one-to-one correspondence is a transformation of the plane.
Because transformations are functions, they have properties of functions.
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4. Properties of Translations
A translation is a motion of a plane that moves every point of the plane a specified distance in a specified direction along a straight line.
A figure and its image are congruent.
The image of a line is a line parallel to it.
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5. Constructions of Translations
To construct the image of an object under a translation, we first need to know how to construct the image A′ of a single point A.
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6. Constructions of Translations
To construct the translation image A′ of a point A with a compass and straightedge, construct a parallelogram MAA′N so that AA′ is in the same direction as MN.
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Example 13-1
Find the image of AB under the translation from X to X′ pictured on the dot paper.
The image of each point on the dot paper can be obtained by first shifting it 2 units down and then 2 units to the right.
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8. Coordinate Representation of Translations
Point (x, y)
Image Point (x + 5, y – 2)
A(–2, 6)
A′(3, 4)
B(0, 8)
B′(5, 6)
C(2, 3)
C′(7, 1)
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9. Translation in a Coordinate System
A translation is a function from the plane to the plane such that to every point (x, y) corresponds the point (x + a, y + b), where a and b are real numbers.
This can be written symbolically as
(x, y) → (x + a, y + b)
where “→” denotes “moves to.”
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Example 13-2
Find the coordinates of the image of the vertices of quadrilateral ABCD under the translations in parts (a) through (c).
a. (x, y) → (x − 2, y + 4)
A(0, 0) → A′(−2, 4)
B(2, 2) → B′(0, 6)
C(4, 0) → C′(2, 4)
D(2, −2) → D′(0, 2)
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Example (continued)
b. A translation determined by the slide arrow from A(0, 0) to A′(−2, 4).
This is the same translation as in part (a).
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Example (continued)
c. A translation determined by the slide arrow from S(4, −3) to S′(2, 1).
The translation from S(4, −3) to S′(2, 1) moves an x-coordinate −2 units and a y-coordinate 4 units.
Thus, every point (x, y) moves to a point with coordinates (x − 2, y + 4). This is the same translation as in part (a).
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Example (continued)
The square ABCD and its image A′B′C′D′ are shown here.
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Rotations
A rotation is a transformation of the plane determined by holding one point, the center, fixed and rotating the plane about this point by a certain amount in a certain direction (a certain number of degrees either clockwise or counterclockwise).
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Rotations
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Isometries
It can be shown that under any isometry, the image of a line is a line, the image of a circle is a circle, and the images of parallel lines are parallel lines.
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Example 13-3
Find the image of ΔABC under the rotation shown with center O.
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Special Rotations
A rotation of 360° about a point moves any point and hence any figure onto itself. Such a transformation is an identity transformation. Any point may be the center of such a rotation.
A rotation of 180° about a point is a half-turn.
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19. Slopes of Perpendicular Lines
We can use transformations to investigate various mathematical relationships, including the slopes between two non-vertical perpendicular lines.
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20. Slopes of Perpendicular Lines
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21. Slopes of Perpendicular Lines
Two lines, neither of which is vertical, are perpendicular if, and only if, their slopes m1 and m2 satisfy the condition m1 m2 = −1. Every vertical line has no slope but is perpendicular to a line with slope 0.
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Example 13-4
Find the equation of line ℓ through point (−1, 2) and perpendicular to the line y = 3x + 5.
The line y = 3x + 5 has slope 3, so the line ℓ has
slope
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Example (continued)
The equation of line ℓ is
The equation of line ℓ is
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