4-1 congruence and transformations
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Transcript 4-1 congruence and transformations
4-1 Congruence and transformations
SAT Problem of the day
Objectives
Draw, identify, and describe transformations in the
coordinate plane.
Use properties of rigid motions to determine whether
figures are congruent and to prove figures congruent.
Transformations
What is a transformation?
Answer: is a change in the position, shape, or size of a
figure.
What are some types of transformations?
Answer: translations(slides),reflections, rotations and
dilations
Example#1
Apply the transformation M to the polygon with the
given vertices. Identify and describe the
transformation.
A. M: (x, y) → (x - 4, y + 1)
P(1, 3), Q(1, 1), R(4, 1)
translation 4 units left and 1 unit up
Example#2
B. M: (x, y) → (x, -y)
A(1, 2), B(4, 2), C(3, 1)
reflection across x-axis
Example#3
. M: (x, y) → (y, -x)
R(-3, 0), E(-3, 3), C(-1, 3), T(-1, 0)
90°rotation clockwise with center of rotation
(0, 0)
Example#4
. M: (x, y) → (3x, 3y)
K(-2, -1), L(1, -1), N(1, -2))
dilation with scale factor 3 and center (0, 0)
Student guided practice
Do problems 3 -6 in your book page 220
Types of transformations
What is isometry ?
An isometry is a transformation that preserves length,
angle measure, and area. Because of these properties, an
isometry produces an image that is congruent to the
preimage.
What is a rigid transformation?
A rigid transformation is another name for an isometry.
Transformations and congruence
Example#5
Determine whether the polygons with the given
vertices are congruent.
.
A(-3, 1), B(2, 3), C(1, 1)
P(-4, -2), Q(1, 0), R(0, -2)
The triangle are congruent; △ ABC can be
mapped to △PQR by a translation: (x, y) →
(x - 1, y - 3).
Example#6
B. A(2, -2), B(4, -2), C(4, -4)
P(3, -3), Q(6, -3), R(6, -6).
The triangles are not congruent; △ ABC can
be mapped to △ PQR by a dilation with scale
factor k ≠ 1: (x, y) → (1.5x, 1.5y).
Student guided practice
Do problems 7 and 8 in your book page 220
Example#7
Prove that the polygons with the given vertices are
congruent.
A(1, 2), B(2, 1), C(4, 2)
P(-3, -2), Q(-2, -1), R(-3, 1)
△ ABC can be mapped to △ A′B′C′ by a
translation: (x, y) → (x – 3, y + 1); and then △
A′B′C′ can be mapped to △PQR by a
rotation: (x, y) → (–y, x).
Example#8
Prove that the polygons with the given vertices are
congruent: A(-4, -2), B(-2, 1), C( 2, -2) and P(1, 0),
Q(3, -3), R(3, 0).
The triangles are congruent because ABC can be
mapped to A’B’C’ by a translation (x, y) → (x + 5, y +
2); and then A’B’C’ can be mapped to ABC by a
reflection across the x-axis
Student guided practice
Do problemsd9 and 10 in your book page 220
Architecture question?
Is there another transformation that can be used
to create this frieze pattern? Explain your answer.
Repeated reflections can create this frieze
pattern; a reflection of any section over a
line through either the left or right side of
each section.
Homework!!!
Do problems even problems 13-24 in your book page
220
Have a great day!!!