Download Classwork 84H TEACHER NOTES Perform Reflections Using Line

Classwork 84H
Perform Reflections Using Line of Symmetry or Matrix
CRS
Objective
TEACHER NOTES
GEOMETRY CONTENT
Perform reflections using line of symmetry in the coordinate plane or by using a matrix
Homework Quiz (Project):
1) ABC with vertices A(–2, 4), B(6, 2),
and C(3, –2) is translated to A'B’C'.
Determine the translation using a
vector in component form, and
determine the coordinates of the
remaining vertices.
2)
B'(8, 6)
3) ABC has vertices A(2, -1), B(1, 3), and C(-2,-2). Write a matrix equation that
shows how to find the image matrix that represents the translation of ABC 4
units to the left and 5 units up.
*hint:
Translation Matrix + ABC Matrix = A’B’C’ Matrix
Homework Quiz (Answers):
1) ABC with vertices A(–2, 4), B(6, 2), and C(3, –2) is
translated to A'B’C'. Determine the translation using a
vector in component form, and determine the
coordinates of the remaining vertices.
2)
B'(8, 6)
3) ABC has vertices A(2, -1), B(1, 3), and C(-2,-2). Write a matrix equation that shows how to find the image matrix that
represents the translation of ABC 4 units to the left and 5 units up.
In Notebooks:
Last week we translated figures using a rule in the coordinate plane or by using a matrix. Today we will learn
how to reflect a figure in the coordinate plane using a line of symmetry and by using a matrix.
In a reflection, the “mirror line” is called the line of reflection. A reflection is an isometry because it preserves
side lengths and angle measures.
Ex:
 Go to CW and go through “Common lines of reflection in the coordinate plane” (#1 – 4)
Using matrices to reflect figures
We can also use matrices to reflect figures… Let’s use segment AB in the coordinate plane to be A(1, 3) and
B(2, 4). Let’s see what happens when we multiply the matrix for segment AB by the following:
1.) Multiply:
1 0 1 2 

 

0 1 3 4 
1 0 

is called the identity matrix (think, it’s like multiplying any # by 1 and you get the same answer back. If
0 1 
you multiply any matrix by the identity matrix, you will get the same matrix back!). So, this has NO EFFECT on
reflecting segment AB in the coordinate plane.
2.) Multiply:
1 0  1 2 

 

0 1 3 4 
Since the sign changes on only the y-coordinate of our coordinate points, this reflects the image over the xaxis:
3.) Multiply:
1 0 1 2 

 

0 1 3 4 
Since the sign changes on only the x-coordinate of our coordinate points, this reflects the image over the yaxis:
FUN CHALLENGE: What matrix will reflect a figure over the line y = x? y = -x?
*Check exit slip on CW
Classwork 84H
Perform Reflections Using Line of Symmetry or Matrix
Name: ________________________
Date: ______________ Pr: _______
Common lines of reflection in the coordinate plane:
1) x-axis (or x = real #)
2) y-axis (or y = real #)
Example 1: The vertices of Δ ABC are A(1, 2), B(3, 0), and
C(5, 3). Graph the reflection of
ΔABC described in the line n: x = 2
Example 2: The vertices of Δ ABC are A(1, 2), B(3, 0), and
C(5, 3). Graph the reflection of
ΔABC described in the line n: y = 3
a. Graph:
a. Graph:
b. What happens to the coordinate points when reflected
over a vertical line?
b. What happens to the coordinate points when reflected
over a horizontal line?
3) the line y = x
4) the line y = -x
Example 3: The endpoints of CD are C(–2, 2) and D(1, 2).
Reflect the segment in the line y = x. Graph the segment
and its image.
Example 4: The endpoints of AB are A(–1, 3) and B(2, 1).
Reflect the segment in the line y = x. Graph the segment
and its image.
a. Graph:
a. Graph:
b. What happens to the coordinate points when reflected
over the line y = x?
b. What happens to the coordinate points when reflected
over the line y = -x?
Example 5: The vertices of Δ DEF are D (1, 2), E (2, 3), and F (4,1). Find the reflection of Δ DEF in the y-axis
using matrix multiplication. Graph Δ DEF and its image.
Step 1: Multiply the polygon matrix by the matrix for a
reflection in the y-axis.
Step 2: Graph Δ DEF and Δ D’E’F’.
TEXTBOOK PAGE _____ #3 – 18 (You will need graph paper!)
Exit Slip (do NOT start until I tell you to!)
1) The endpoints of JK are J(–1, –2) and K(1 –2).
Reflect the segment in the given line. Graph the
segment and its image.
a. y = 1
2) The vertices of Δ QRS are Q (–1, 4), R (0, 1), and
S (2, 3).
a. Find the reflection of ΔQRS in the x-axis using
matrix multiplication.
b. Graph Δ QRS and its image.
b. y = -x
Exit Slip (ANS KEY)
1) The endpoints of JK are J(–1, –2) and K(1 –2).
Reflect the segment in the given line. Graph the
segment and its image.
a. y = 1
2) The vertices of Δ QRS are Q (–1, 4), R (0, 1), and
S (2, 3).
a. Find the reflection of ΔQRS in the x-axis using
matrix multiplication.
b. Graph Δ QRS and its image.
b. y = -x