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Lecture 2

MA 108 Lecture Notes - Lecture 2: Bisection, Pentagon, Quadrilateral


Department
Mathematics
Course Code
MA 108
Professor
Elyse Suzanne Rogers
Lecture
2

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3) Reflection in a Line: A reflection in a line m maps a preimage point P in the plane to an
image point P’ so that the following properties are true:
1) If P is not on line m, then line m is the perpendicular bisector of segment PP’.
2) If P is on line m, then P = P’.
Line m is called the line of reflection. A reflection is an isometry, so reflections preserve size
(and shape). Reflections reverse the orientation of figures.
While reflection is a quite general concept and the line of reflection can be anywhere in
the plane, we will concentrate on four specific lines of reflection in the coordinate plane: the x-
axis, the y-axis, the line y = x, and the line y = −x.
Case 1: Reflection in the x-axis. When a point is reflected in the x-axis, its x-coordinate stays
the same, while its y-coordinate changes sign. In coordinate notation, (x’, y’) = ( x, −y ). For
example, the point ( 3, 5) has the image (3, -5), and the point ( -2, -7) has the image ( -2, 7).
Any point on the x-axis is its own image.
If a figure is represented as a polygon matrix, then it can be reflected in the x-axis by
multiplying the polygon matrix by the reflection matrix [ 1 0 . The reflection matrix goes in
front
0 -1 ]
of the polygon matrix (this is called “premultiplying”). For example, suppose we want to reflect
triangle ABC in the x-axis, where A = ( 1, 6), B = ( 3, 8), and C = ( 5, 4). The matrix
multiplication and image look like this:
[ 1 0 [ 1 3 5 = [ 1 3 5
0 -1 ] 6 8 4 ] -6 -8 -4 ]
Try these: Find the reflection of each point in the x-axis:
1) ( 9, 4) (9, -4)
2) (-4, 7) ( -4,-7)
3) ( 2, -5) ( 2, 5)
4) ( -5, -3) (-5, 3)
Write the matrix equation to reflect triangle ABC [A = ( 0, -2), B = ( 6, 1), C = ( 4, 3)] in the x-
axis:
[ 1 0 [ 0 6 4 = [ 0 6 4
0 -1 ] -2 1 3 ] 2 -1 -3 ]
Case 2: Reflection in the y-axis. When a point is reflected in the y-axis, its y-coordinate stays
the same, while its x-coordinate changes sign. In coordinate notation, (x’, y’) = ( −x, y ). For
example, the point ( 3, 5) has the image (-3, 5), and the point ( -2, -7) has the image ( 2, -7).
Any point on the y-axis is its own image.
If a figure is represented as a polygon matrix, then it can be reflected in the y-axis by
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multiplying the polygon matrix by the reflection matrix [ -1 0 . The reflection matrix goes in
0 1 ]
front of the polygon matrix. For example, suppose we want to reflect triangle ABC in the y-axis,
where A = ( 1, 6), B = ( 3, 8), and C = ( 5, 4). The matrix multiplication and image look like this:
[ -1 0 [ 1 3 5 = [ -1 -3 -5
0 1 ] 6 8 4 ] 6 8 4 ]
Try these: Find the reflection of each point in the y-axis:
1) ( 9, 4) (-9, 4)
2) (-4, 7) ( 4, 7)
3) ( 2, -5) (-2, -5)
4) ( -5, -3) (5, -3)
Write the matrix equation to reflect triangle ABC [A = ( 0, -2), B = ( 6, 1), C = ( 4, 3)] in the y-
axis:
[ -1 0 [ 0 6 4 = [ 0 -6 -4
0 1 ] -2 1 3 ] -2 1 3 ]
Case 3: Reflection in the line y = x. When a point is reflected in the line y = x, its coordinates
are interchanged. The original x-coordinate becomes the y-coordinate of the image, and the
original y-coordinate becomes the x-coordinate of the image. In coordinate notation, (x’, y’) =
( y, x ). For example, the point ( 3, 5) has the image ( 5, 3 ), and the point ( -2, -7) has the
image ( -7, -2). Any point on the line y = x is its own image.
If a figure is represented as a polygon matrix, then it can be reflected in the line y = x by
multiplying the polygon matrix by the reflection matrix [ 0 1 . The reflection matrix goes in
front
1 0 ]
of the polygon matrix. For example, suppose we want to reflect triangle ABC in the line y = x,
where A = ( 1, 6), B = ( 3, 8), and C = ( 5, 4). The matrix multiplication and image look like this:
[ 0 1 [ 1 3 5 = [ 6 8 4
1 0 ] 6 8 4 ] 1 3 5 ]
Try these: Find the reflection of each point in the line y = x:
1) ( 9, 4) ( 4, 9 )
2) (-4, 7) ( 7, -4 )
3) ( 2, -5) ( -5, 2 )
4) ( -5, -3) ( -3, -5 )
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