13.4 Compositions
POINT SYMMETRY: A simple test to determine whether a figure has point symmetry is to turn it upside­
down and see if it looks the same. A figure that has point symmetry is unchanged in appearance by a 180 degree rotation.
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ROTATIONAL SYMMETRY: the degree measure of the smallest angle needed to rotate a regular polygon of n sides to be its own image is 360/n
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LINE SYMMETRY, or just symmetry, occurs when two halves of a figure mirror each other across a line. The line of symmetry is the line that divides the figure into two mirror images. A simple test to determine if a figure has line symmetry is to fold the figure along the supposed line of symmetry and see if the two halves of the figure coincide. 3
***Certain figures, such as the square and the circle, possess all 3 types of symmetry.***
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PUTTING IT ALL TOGETHER …
When two or more transformations are combined to form a new transformation, the result is called a COMPOSTION of transformations. The symbol for a composition of transformations is an open circle. o
rx‐axis o T3,4(x,y)
The notation is read as a reflection in the x‐axis following a translation of (x+3, y+4). Be careful!!! The process is done in reverse!!
rx‐axis o T3,4(x,y)
1st: translation of (x+3,y+4)
2nd: reflect your new point over the x­axis
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In the accompanying figure, k and n are lines or symmetry for square ABCD.
Find rn o rk(A)
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In the accompanying diagram, p and q are lines of symmetry for square ABCD, and M is the midpoint of diagonal AC. Find Rm o rp o rq (D)
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In square ABCD, p and n are lines of symmetry, and M is the midpoint of AC. What is rn o Rm,90 o rp(AB)?
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13.5 ‐ Composition Rules
Does order matter?
T(3,0) o D4 (x,y)
D4 o T(3,0)(x,y)
Composition of Transformations is NOT commutitive
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Glide Relfection: the compostition of a line reflection and a translation where order doesn't matter
EX)
Draw and label coordinates A(2,‐6), B(2,‐2) and C(4,‐5)
a) find the coordinates of A'B'C', the image of ABC after a translation
of T0,7
b) find the coordinates of A"B"C",
the image of A'B'C' after a reflection
in the y‐axis
c) Name the single trasnformation the maps triangle ABC onto triangle A"B"C"
1) line reflection 2) glide reflection 3) point reflection 4) rotation 10
Triangle ABC has vertices A(0,3), B(2,3) and C(4,5)
a) graph ABC and its image
A'B'C' after a reflection in the
the line y=x
b) graph A"B"C, the image
of A'B'C' after a translation
of T‐3,3
c) name the single transformation
that maps triangle ABC onto
triangle A"B"C"
d) find the rule of the transformation equivilant to T‐3,3 o ry=x
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Using the diagram, the equation of line k is y=‐x, the equation of the line m is y=2, and point A=(‐3,‐2). Find the coordinates of rm o rk(A)
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Name the single transformation that is equivalent to the composition rx‐axis o ry=x
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Homework: 13.4 pg 582 #1‐4,11,12,17,20,23,30,33,34,41‐44,49,51,59
13.5 pg 588 #3,4,9,17,19,22,23
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