7.1 Rigid Motion in a Plane (396)
Notes #2-21
Date: ______
A
W.1 Which segment is the longest? Why?
C
B
70°
60°
60°
65° 60° 50°
80°
E
55° 40° D
Transformation: an operation that maps (moves) a preimage onto an image.
Orientation: counterclockwise: up from x-axis (+) or clockwise: down (-)
Ex.1 a. Name and describe the transformation.
b. Name the coordinates of the vertices of
the image.
c. Is ΔDEF congruent to its image?
d. What orientation does each figure have?
ΔDEF
ΔD’E’F’
Isometry (rigid transformation): a transformation that preserves length,
angle measures, parallel lines and distances between points.
Ex.2 Which of the following transformations appear to be isometries?
1 Ex.3 ΔGHJ is mapped onto ΔUVW. The mapping is a translation. Given
ΔGHJ→ΔUVW is an isometry, UV= 8, and mH= 37°, find the length
of GH and mV.
Ex.4 The blades on a windmill are titled to catch
wind, which then turns them.
a. How are blades A and B related?
b. Explain how knowing how the blades are
related helps build the windmill.
Ex.5 ΔABC is reflected over the x-axis. Its reflection is ΔA’B’C’. If the
coordinates of A are (3, 2), what are the coordinates of A’?
R.1
The nth term of a sequence is 3n + 2. The current term is 20.
What is the next term?
A.
B.
C.
D.
21
23
27
30
2 7.2 Reflections (404)
W.1 A(7, 7), B(2, 3) & C(7, -2). Which coordinates
of D result in a parallelogram?
A.
B.
C.
D.
(2, -5)
(2, 12)
(12, 1)
(16, 3)
Notes #2-22
Date: ______
10
5
5
10
-5
Ex.1 Graph the given reflection.
a. ΔUVW in the x-axis.
U (-3, 3)
U’ _______
V (-3, 0)
V’ _______
W (2, 1)
W’ _______
b. What orientation does each figure have?
ΔUVW
ΔU’V’W’
c. Z(1, -7) in the line y = x. Z’ _______
What are the coordinates of the image of (x, y) when it is reflected in the
a) x-axis?
b) y-axis?
Reflection Theorem (7.1) A reflection is an isometry.
3 Ex.2
Line of symmetry: a line that a figure can be reflected in and be mapped onto itself.
Ex.3 Determine the number of lines of symmetry in each quadrilateral.
a.
b.
c.
R.1 In ΔMNP, the length of side MN is 5 units. M (3,6) & N (x, 10).
Which is a possible value of x?
A. -2
B. -1
C. 0
D. 1
R.2 In isosceles Δ PQR, P is the vertex angle. If mQ = 8x – 3
and mR = 2x + 15, what is mP?
A. 3°
B. 21°
C. 42°
D. 138°
4 7.3 Rotations (412)
Notes #2-23
Date: ______
Rotation Theorem (7.2) A rotation is an isometry.
Ex.1 Given: A rotation about P maps Q onto Q’ and R onto R’.
Prove: Q  Q’
Statements
Reasons
1. A rotation about P maps Q onto Q’ and R onto R’. 1. Given
2. PQ
= PQ’, PR = PR’
3. QR
4.
2. Def of rotation
= Q’R’
5. ΔPQR
3. Rotation
is an isometry
4.
 ΔPQ’R’
5. SSS
6. Q  Q’
6. CPCTC
Ex.2 A quadrilateral has vertices P(3,-1), Q(4,0), R(4,3), and S(2,4). Rotate
PQRS counterclockwise about (0,0) and
name the coordinates of the new vertices.
a) 180°
P’( , )
Q’( , )
R’(
,
)
S’(
,
)
R”(
,
)
b) 270°
P”(
,
)
Q”(
,
)
c) What orientation does each figure have?
PQRS
P’Q’R’S’
S”(
,
)
P”Q”R”S”
5 Ex.3 ΔJKL is reflected in line k to
produce ΔJ’ K’ L’. This triangle
is then reflected in the line m to
produce ΔJ’’ K’’ L’’. Describe the
transformation that maps ΔJKL
to ΔJ’’ K’’ L’’.
Ex.4 Which figures have rotational symmetry? For those that do, describe
the rotations that map the figure onto itself.
c. Regular Pentagon
Ex.5 Explain how the design can be mapped onto itself by a rotation.
6 7.4 Translations and Vectors (421)
Notes #2-24
Date: ______
Translation Theorem (7.4) A translation is an isometry.
Ex.1 A reflection in line k maps ΔXYZ to
ΔX’Y’Z’, a reflection in line m maps
ΔX’Y’Z’ to ΔX’’Y’’Z’’, k || m, AZ’= 3
and Z’B= 2.
a. Name some congruent segments.
b. What kind of figure is ZZ’’EX? What is the length of ZZ’’?
c. Name a pair of perpendicular segments.
Coordinate notation: (x, y) → (x + a, y + b) shifts horizontally a & vertically b units
7 Ex.2 Sketch a parallelogram with vertices
R(-4,-1), S(-2,0), T(-1,3), U(-3,2). Then
sketch the image of the parallelogram
after translation (x, y) → (x + 4, y – 2).
R’
S’
T’
U’
What orientation does each figure have?
RSTU
R’S’T’U’
Vector: a quantity that has both direction and magnitude, symbol: arrow
Ex.3 In the diagram, name each vector and write its component form.
Ex.4 The component of vector RS is 2,-3 . Use vector RS to translate the
quadrilateral whose vertices are:
G(-3, 5), H(0, 3), J(1, 3), and K(2, 5).
8 Ex.5 ΔABC → ΔA’B’C’ using translation. A(-4,5), B(-1,-1), and C(2,3).
A’(-3,2), B’(0,-4), and C’(3,0). Write the component form of the
vector that describes the translation.
Ex.6 The coordinates of a logging site are S(5, 7). A truck traveling in a
straight line from the site to a mill at M encounters a road detour at
D when it is 4 mi west and 2 mi south of the logging site. The truck
must travel an alternate route to A(4, 3).
a. Write the component forms of the
vectors from S to D and from D to A.
b. The mill is 6 mi west and 3 mi south of
the logging site. Write the component
form of the vector that describes the
route the logging truck can follow to
arrive at the mill.
Derive the quadratic formula from: ax2 + bx + c = 0.
9 7.5 Glide Reflections and Compositions (430)
Notes #2-25
Date: ______
In the diagram below, AD and BC bisect each other at E.
Which congruence postulate or theorem would prove these
two triangles are congruent?
A
C
E angle-angle-angle
angle-side-angle
B
D
side-angle-side
side-side-side
Glide reflection: a transformation in which
1. A translation maps P to P’.
2. A reflection in a line k parallel to the
direction of the translation maps P to P”.
Or a reflection and then a translation.
Composition Theorem (7.6) The composition of two (or more) isometries is
an isometry.
Ex.1 Use the information to sketch the
image of ΔQRS after a glide reflection.
Q(2,-3), R(4,-4), and S(5,-1)
Translation: (x, y) → (x, y+5)
Reflection: in the y-axis.
Q’(
,
)
R’(
,
)
S’(
,
)
10 Ex.2 Sketch the image of CD after a
composition of the given rotation and
reflection. C(2,0), D(3,3)
Reflection: in the x-axis
Rotation: 270° counterclockwise
about the origin.
C’(
,
)
D’(
,
)
Ex.3 Repeat example 2, but switch the order
of the composition by performing the
rotation first and the reflection second.
C(2,0), D(3,3). What do you notice?
C’(
,
)
D’(
,
)
Ex.4 Describe the composition of
transformations in the diagram.
Ex.5 Describe the composition of
transformations in the diagram.
11