Regents Examination in Geometry: Transformations and Congruence

Transformations and Congruence

Rigid Motion and Similarity Transformations

Key Ideas
A geometric transformation is a function that operates on a figure and may change its position, size, shape, or orientation. The figure can be any geometric figure, picture, or even a point. The input of the function, or original figure, is called the preimage. The output of the function is called the image. Every point of the preimage is mapped to a corresponding point of the image by the transformation. The transformation is defined by the rule that describes the relationship between the preimage and the image. A transformation in which the distance between any two points is preserved is called a rigid motion. Rigid motions result in an image congruent to the preimage. A transformation in which the shape, but not necessarily the size, is preserved is called a similarity transformation. The named rigid motions studied in this course are translations, rotations, and reflections. The one named similarity transformation we will study is the dilation. The rigid motion transformation will be used to define and determine congruence between a preimage and an image.


Transformation Definitions and Notation
A transformation is a function that takes an input called the preimage and returns an output called the image. The preimage and image can be points, segments, or any geometric figure or picture. We say the transformation maps the preimage onto the image. Figure 5.1 shows an example of a transformation that maps △ABC to △A′B′C′. △ABC is the preimage, and △A′B′C′ is the image. The transformation operates on the preimage point by point so that every point in the preimage has a corresponding point in the image.

Transformation of △ABC to △A′B′C′


Math Facts
Transformations in geometry are similar to one-to-one functions in algebra. One-to-one functions have only one possible y-value for each x-value, and only one possible x-value for each y-value. With transformations, each point in the preimage has only one corresponding point in the image. Also, given a point in the image there is only possible point in the preimage that could have led there.

The transformation rule can be specified in different ways. It can be given in terms of a geometric relationship between points in the image and preimage or as an algebraic rule that operates on the coordinates of each point in the coordinate plane.
When naming corresponding pairs of points, the same letter is used, with the single prime symbol ′ used to indicate the transformed point. An arrow, , can also be used to indicate the operation of a transformation, as in △ △A′B′C′.

Rigid Motion and Invariance
A transformation in which the distance between any two points is unchanged is called a rigid motion or isometry. In other words, the lengths of any corresponding segments are equal. The image formed after a rigid motion will always be congruent to the preimage, and corresponding angles will also be congruent.
The transformation shown in Figure 5.2 is a rigid motion with congruent corresponding lengths: AB ≅ A′B′, BC ≅ B′C′, CD ≅ C′D′, DA ≅ D′A′, AC ≅ A′C′, and BD ≅ B′D′. Corresponding angles are also congruent: ∠A ≅ ∠A′, ∠B ≅ ∠B′, ∠C ≅ ∠C′, and ∠D ≅ ∠D′. The image is held rigid in size and shape but not necessarily in position or orientation.

A rigid motion transformation

Note that the distance between all corresponding points must be the same. In Figure 5.2, simply requiring the four sides of rectangle ABCD to maintain their length is not sufficient to ensure rigid motion. Parallelogram ABCD with angles not equal to 90° could still have AB ≅ A′B′, BC ≅ B′C′, CD ≅ C′D′, and DA ≅ D′A′.

Quantities or properties that remain unchanged after a transformation are described as invariant or preserved.

In the Figure , side lengths, angle measures, and area are all invariant. Slope is not invariant since the slopes of corresponding sides
 and  are not equal. Note, however, that parallelism is invariant because any pair of sides that are parallel in the preimage will still be parallel in the image.

Some properties that remain invariant under any rigid motion include:
Length
Angle
Parallelism
Area

Figure below illustrates three specific types of transformations that are rigid motions. Each will be discussed in more detail.

Rigid motion transformations

Similarity Transformations
A transformation in which all corresponding lengths are in the same ratio and all corresponding angles are congruent is called a similarity
transformation. The basic similarity transformation we will discuss is the dilation. Figure 5.4 illustrates a dilation with a ratio, or scale factor of

Dilations can be thought of as an enlargement or a reduction of the preimage. The preimage and image after a similarity transformation are not necessarily congruent; they are congruent only if the scale factor equals 1.

However, corresponding angles will always be preserved. The result is an image that has the same shape as the preimage but not necessarily the same size.

Dilation with ratio equal to 

Orientation
Orientation of a polygon refers to the direction traveled around a figure when moving along consecutive vertices. The orientation can be clockwise or counterclockwise. In the transformation illustrated in Figure 5.1, the orientation of the preimage is clockwise since one would travel in a clockwise direction to go from vertex A to B to C. However, the orientation of the image is counterclockwise since that is the direction of travel from vertex A′ to B′ to C′.
The transformation shown in Figure 5.2 preserves orientation. In both the image and preimage, one travels counterclockwise from vertex A to B to C to D and also from A′ to B′ to C′ to D′.


Math Facts
A rigid motion or isometry that preserves orientation is called direct. A rigid motion or isometry that does not preserve orientation is called opposite.
Translation—direct rigid motion
Rotation—direct rigid motion
Reflection—opposite rigid motion

Properties of Transformations

Reflections
A line reflection flips a figure over a line so that the figure appears as a mirror image. The line is called the line of reflection. The notation rℓ indicates a reflection over line ℓ.
The line of reflection is always the perpendicular bisector of the segment joining corresponding points in the preimage and image. As a result, every point on the preimage and its corresponding point on the image are equidistant from the line of reflection.
Figure 5.5 illustrates the reflection of point A over line ℓ. Note that line ℓ is the perpendicular bisector of


Reflection of a point over a line

Figure shows the reflection of a quadrilateral. Note that the orientation has changed from clockwise to counterclockwise. A single reflection will always change the orientation.

Reflection of quadrilateral ABCD


Math Facts
In every reflection, the segments constructed through pairs of corresponding points are parallel.

In the Figure, . The midpoints of each of these segments also lie on the line of reflection. Reflections preserve distance and angle, so they are isometries. However, orientation is not preserved.


Example 1
△F′G′H′ is the image of △FGH after a reflection over line  as shown in the figure. Classify quadrilateral GG′H′H. Justify your answer.

Solution:
 because segments through corresponding pairs of points are parallel.
 because length is preserved. Therefore GG′H′H must be an isosceles trapezoid.

Example 2
Prove segments connecting corresponding points after a reflection are parallel.

Given:
 is the reflection of  over line m
Prove:

Solution:

Statements	Reasons
is the reflection of over line m	Given
⊥ line m, ⊥ line m	The line of reflection is the perpendicular bisector of the segment joining corresponding points
	Segments perpendicular to the same line are parallel

Translations
A translation is a transformation that simply slides, or moves, a figure from one position to another. Translations can be specified using a vector with the notation 
Figure shows pentagon ABCDE translated to A′B′C′D′E′ by vector . A vector consists of a magnitude and a direction, as illustrated by vector
. The magnitude is represented by the length of the vector. In this case, the length is FG. The direction is indicated by following the endpoint, point F, toward the arrow. Pentagon ABCDE is translated a distance of FG parallel to the direction of .

Vector definition of a translation

Math Facts
In every translation, the segments constructed through pairs of corresponding points are both parallel and congruent. For example in Figure 5.7, AA′ || BB′ || CC′ || DD′ || EE′. Also, AA′ ≅ BB′ ≅ CC′ ≅ DD′ ≅ EE′. Translations preserve distances and angles, so translations are an isometry. They also preserve parallelism and slope.


Example 1
Segment  is translated by vector  to , MN = 4, and FG = 6. What can you conclude about the lengths of M′N′, MM′, and NN′? Explain your reasoning.
Solution:
MN = M′N′ = 4 because translations are isometries that preserve length. MM′ = NN′ = 6 because every point on  is translated by the distance of , which is 6.

Example 2
In example 1 above, classify quadrilateral MNN′M′. Justify your answer.
because translations preserve parallelism.
 because each point on  is translated in the same direction. Therefore both pairs of opposite sides are parallel. So MNN′M′ must be a parallelogram.

Example 3
Given:
 is the translation of  by vector 

Prove: m∠BAC + m∠ACD = 180°
Solution:

Statements	Reasons
	Given
	The image of a segment after a translation is parallel to the preimage
∠BAC and ∠ACD are supplementary	Same side interior angles formed by parallel lines are supplementary
m∠BAC + m∠ACD = 180°	The measures of supplementary angles sum to 180°

Rotations
A rotation is the spinning of a figure about a pivot point called the center of rotation. The angle of rotation is measured counterclockwise unless otherwise specified. For example, Figure 5.8 shows point P rotated 60° to point P′ about the center of rotation at point O. The notation RC,a is used to specify a rotation, where a is the angle of rotation and C is the center of rotation.

Rotation of a figure about a point


Math Facts
In a rotation, the angle of rotation found using any pair of corresponding points in the preimage and image will be congruent. In Figure below, ∠AOA′ ≅ ∠BOB′ ≅ ∠COC′ ≅ ∠DOD′. In addition, the distances from each of a corresponding pair of points to the center of rotation are equal. In the Figure,

,
,
, and
.

Rotation of trapezoid ABCD
 

Example 1
The accompanying figure could represent which transformations?


translation or rotation
translation or reflection
reflection or rotation
reflection or dilation

Solution: Choice (3) is correct. The figure could represent a rotation about point A or a reflection over line .


Example 2
D′E′F′G′H′ is the image of DEFGH after a rotation about point O. If m∠DOD′ = 70°, find m∠OGG′.

Solution: Since the figure represents a rotation, m∠GOG′ = 70° and . Applying the isosceles triangle theorem to △GOG′, m∠OGG′ is calculated as:


Dilations
A dilation enlarges or reduces the size of a figure without changing its shape. All angles in the image remain congruent. However, the lengths of segments in the image are proportional to lengths in the preimage. The ratio of lengths is called the scale factor. A dilation is not a rigid motion since the image may not be congruent to the preimage. Dilations are a type of similarity transformation because the image is always similar to the preimage. Similar figures have the same shape but may differ in size. 
Dilations are specified about a center point. DC,r is a dilation with a center point of C and a scale factor of r. It maps point P onto point P′ such that P′ lies on the ray 

and .

Dilation  of about C with scale factor 2

Transformations in the Coordinate Plane

Key Ideas
Transformations can map one point to another on the coordinate plane. Specific rules exist for finding the new coordinates given the coordinates of the preimage.

Reflections Over a Line in the Coordinate Plane
To reflect a point over any horizontal or vertical line, flip the point over the line so the distances from point to line are equal. If the line is horizontal or vertical, this can be done by counting grid units on a graph as shown in Figure below. Point B is 2 units to the left of the line x = −1. So after rx=−1, point B′ is located 2 units to the right of x = −1. The same is done for point A.

Reflection of  over x = −1

Algebraically, the change in the x- or y-coordinate between the preimage and the line of reflection can be calculated and applied to locate the image on the other side of a horizontal or vertical line of reflection.

A set of rules can also be applied to the special cases of reflections in the coordinate axes, y = x or y = −x:
rx-axis(x, y) → (x, −y)
ry-axis(x, y) → (−x, y)
ry=x(x, y) → (y, x)
ry=−x(x, y) → (−y, −x)

Example
Point R(7, 3) is reflected over the line y = x. Find the coordinates of the image R′.
Solution: The rule for a reflection over the line y = x is (x, y) → (y, x), so we switch the two coordinates. R′ has coordinates (3, 7).

Reflections Through a Point in the Coordinate Plane
The coordinates of a point after reflection through a point P can be calculated by counting the horizontal and vertical distance to P. Figure below illustrates the reflection of △ABC through point P. The image is △A′B′C′.

Reflection through a point

The horizontal distance from A to P is (0 − (−5)) = 5. The vertical distance is (−1 − (−4)) = 3. Since P must be the midpoint of
, A′ must be located the same horizontal and vertical distance from P. So we add those distances to the coordinates of P. The coordinates of A′ are (0 + 5, −1 + 3), or A′(5, 2). The same procedure is repeated to find B′ and C′.
A reflection through a point is also equivalent to a 180° rotation, which has the rule (x, y) → (−y, x). You can confirm this rule with the coordinates shown in Figure 5.12.

Translations in the Coordinate Plane
In the coordinate plane, the notation Th,k or (x, y)→ (x + h, y + k) is used for translations. The variables h and k represent the distance the preimage is translated along the x-axis and y-axis, respectively. The signs of h and k indicate the direction of the slide:

In Figure below, △A′B′C′ is the image of △ABC under the translation T4,−2. The image is translated 4 units right (positive x-direction) and 2 units down (negative y-direction).

Translation of △ABC

The coordinates of the image can be determined by counting grid units from the preimage in the appropriate x- and y-directions or can be calculated algebraically. Use the translation in Figure 5.13 as an example. The coordinates of △A′B′C′ are calculated as follows:

When given a preimage and an image in the coordinate plane, you can determine the translation that was applied by simply counting the distance translated on the graph in each direction. You can also calculate the differences in the x-coordinates and y-coordinates between the two images.

Example 1
The vertices of △MNP have coordinates M(−5, 2), N(1, 2), P(0, 5). Find the coordinates of N′ and P′ if vertex M is mapped to M′(3, 6).
Solution: Solving this problem requires two steps. First determine the translation function. Then apply it to find N′ and P′.

We know that

The translation is therefore T8,4. N′ and P′ can now be found:

As an alternative, the preimage and the image could be graphed and the translation determined graphically. Remember that there is often more than one approach to a problem.

Example 2
After a certain transformation, the image of △BAD with coordinates B(7, 1), A(0, 3), and D(12, 4) is B′(13, 3), A′(6, 5), and D′(18, 1). Is the transformation a translation? Explain why or why not.
Solution: Under a translation, every point in the preimage undergoes the same change in coordinates. If we find the same Th,k for each pair of coordinates, then we know that the entire triangle underwent a translation. Calculate each h and k:

The transformation is not a translation because point D does not undergo the same translation as points B and A. An alternative approach would be to graph the two triangles and show that they are not congruent.

Rotations in the Coordinate Plane
The coordinates after rotations about the origin, Rorigin,a(P) specifies a counterclockwise rotation of point P centered about the origin. If the center of rotation is not specified, it is assumed to be the origin.

The coordinates after a rotation about the origin in increments of 90° are found using the following rules:
R90 (x, y)   = (−y, x)
R180 (x, y) = (−x, −y)
R270 (x, y) = (y, −x)

Multiple rotations of 90° can be applied to achieve rotations of 180° and 270°. (Apply the rotation twice for R180 and 3 times for R270.) After a rotation of 360°, the image is coincident with the preimage and the coordinates remain unchanged.

Math Facts
A graphical alternative to the algebraic rules for rotations is to graph the image and then rotate your paper by the specified rotation. The x-axis is now the y-axis and vice versa. Read the new coordinates, rotate the paper back to the 0° orientation, and plot the new points.

Example
Given △MTV with vertices M(1, 4), T(5, 6), and V(7, 1), graph and label triangles △MTV, △M′T′V′ = R90(△MTV), and △M″T″V″ = R270(△MTV).
Solution:
By applying the rule for R90, we find M′(−4, 1), T′(−6, 5), and V′(−1, 7).
By applying the rule for R270, we find M″(4, −1) T″(6, −5), and V″(1, −7).


Dilations in the Coordinate Plane
To find the image of a point after a dilation about the origin, simply multiply each coordinate by the scale factor.

Example 1
Sketch △ABC with vertices A(2, 1), B(3, 3), and C(0, 2). Sketch and state the coordinates of △A′B′C′, the image after D3(△ABC).
Solution:
Multiply each coordinate by 3.
A(2, 1) → A′(6, 3)      B(3, 3) → B′(9, 9)      C(0, 2) → C′(0, 6)


Example 2
Given  with coordinates A(4, 1) and B(2, 3), find the coordinates of A′ and B′ after D2 centered at C(1, 2).
Solution: The horizontal distance from C to A is 3 units. The vertical distance is −1.
Scaling by a factor of 3 gives 9 units horizontally and −3 units vertically. These are counted from point C:

Repeat for point B. The horizontal distance from C to B is 1 unit. The vertical distance is 1 unit. Scaling by a factor of 3 gives 3 units horizontally and 3 units vertically.

The coordinates are A′(10, −1) and B′(4, 5).

If you are given the graph of a preimage and an image, you can calculate the scale factor in one of two ways:
Find the ratio of two corresponding lengths in the image and preimage.
Find the ratio of the distances to the center of dilation from two corresponding points.

Example
Given  and  which is the image after a dilation about point C, write three different expressions for the scale factor of the dilation. Evaluate the ratio.

Solution:
Using the ratio of corresponding lengths: 
Using the ratio of corresponding distances to the center: 
From point C, D′ is located 2 units left and 6 units up. Point D is located 1 unit left and 3 units up from C. The scale factor is therefore 3. The same value would be obtained using points R and C.


Math Fact
Two examples of transformations that are neither rigid motions nor similarity transformations are the horizontal stretch and the vertical stretch. A horizontal stretch elongates a figure only in the horizontal direction. On the coordinate plane, the x-coordinate of every point is multiplied by a scale factor. A vertical stretch does the same thing to only the y-coordinate.

Symmetry

Key Ideas
A figure has line symmetry if it can be folded in half and every point on one half maps onto a point on the second half. A figure has rotational symmetry if the figure can be rotated by an angle less than 360° and every point on the rotated image maps to a point in the preimage.

Line Symmetry
A figure has line symmetry if it can be reflected over a line that maps one half of the figure exactly onto the other half. The line is called the line of symmetry and divides the figure into two congruent parts. A figure can have zero, one, or more lines of symmetry.

Figures below illustrate shapes with 1, 2, 3, and 0 lines of symmetry.

Figures with 1 line of symmetry

Figures with 2 lines of symmetry

Figure with 5 lines of symmetry

Figures with no lines of symmetry


Math Fact
The number of lines of symmetry in a regular polygon is always the same as the number of sides.


Example 1
How many lines of symmetry are found in a regular hexagon?


1
3
6
12

Solution: 6 lines of symmetry

Example 2
Which of the following letters has more than one line of symmetry?






Solution:




Rotational Symmetry
A figure has rotational symmetry if it can be rotated by some angle 0 < θ < 360° about its center and have every point map to another point on the image. In other words, the image will look identical to the preimage after the rotation. Of course, the image after each of these rotations is congruent to the preimage since rotations are rigid motions.
Figure below shows rectangle ABCD after rotations of 0°, 90°, 180°, 270°, and 360°. Rotations of 0°, 180°, and 360° result in figures that look identical to the original. We say the figure has 180° rotational symmetry. The 0° and 360° rotations are not considered rotational symmetries. Note that the location of individual points has changed. After a 180° rotation, point A is mapped to point C, point B is mapped to point D, point C is mapped to point A, and point D is mapped to point B.

Rectangle ABCD after rotations of 0°, 90°, 180°, 270°, and 360°

Some figures have more than one rotation that results in an identical figure. Figure 5.19 shows 120° and 240° rotational symmetry in an equilateral triangle.

Rotational symmetry in an equilateral triangle

Every figure will look identical after a 0° and 360° rotation about its center. This is why the rotation must be greater than 0° and less than 360° for a figure to have rotational symmetry. Rotations of 0° and 360° are always identity transformations—the image is always identical to the preimage.


Math Fact
A regular polygon with n sides always has rotational symmetry, with rotations in increments equal to its central angle of 360°/n.


Example 1
By how many degrees must a regular octagon be rotated so that it maps onto itself? List all the rotations less than 360° that will map an octagon onto itself.
Solution:

Rotations of 45°, 90°, 135°, 180°, 225°, 270°, and 315°
Although the preimage and image look identical after a symmetry rotation, the position of individual points will differ.

Example 2
By how many degrees must pentagon ABCDE be rotated about its center to map point A to point C?

Solution:

Since rotations are counterclockwise, the pentagon must be rotated three increments of 72°, which equals 216°, to map point A to point C.

Compositions of Rigid Motions

Key Ideas
Any of the transformations can be applied in succession, which is called a composition. The output from the first transformation becomes the input into next. For some compositions of transformations, the order in which they are applied does not matter. For some other compositions, though, the final image depends on the order in which the transformations are applied.

Compositions
A composition of transformations is a sequence of transformations with a specified order. The first transformation is applied to the preimage. The resulting image becomes the preimage of the next transformation. Multiple transformations can be chained together in this way.
Three different notations for compositions may be used. The composition symbol, °, may be used to separate a pair of transformations. The transformation on the right must be done first. Alternatively, transformations may be nested in parentheses. The innermost transformation must be performed first.

Finally, the composition can be stated in words.
 means rotate  90° about point C and then translate it by vector 
 means reflect  over ray  and then dilate through C by a scale factor of 2.

The above two compositions appear in Figure 5.20. The notations A′ and B′ are the result of the first transformation. The notations A″ and B″ are for the second.


Example 1
Which composition of transformations is represented?


RP,180°(rm(△ABC))
rm(RP,180°(△ABC))
rm(rB′C′(△ABC))
RP,90°(rm(△ABC))

Solution: (1) △A′B′C′ is the reflection of △ABC over m, and △A″B″C″ is the rotation of △A′B′C′ by 180°.

Example 2
△RST has coordinates R(6, 2), S(4, −1), and T(2, 3). Graph and label △R″S″T″, the image of △RST after a reflection over the line x = 1 followed by a 180° rotation about the origin. State the coordinates of △R″S″T″.
Solution:

The coordinates are R″(4, −2), S″(2, 1), and T″(0, −3).

For some compositions, the order of transformations does not matter. The final image looks the same after any order of the transformations. For others, the final image will depend on the order specified.

Example 3
Point P has coordinates (5, 1). Does T4,2 ° ry-axis(P) map P to the same point as ry-axis ° T4,2(P)?
Solution:
Using T4,2 ° ry-axis(P): ry-axis(5, 1) → (−5, 1) and T4,2(−5, 1) → (−1, 3)
Using ry-axis ° T4,2(P): T4,2(5, 1) → (9, 3) and ry-axis(9, 3) → (−9, 3)
The compositions map point P to two different points.


Math Fact
Compositions of transformations are similar in concept to transformations of functions you may have studied in algebra. When given two functions, f(x) = x2 and g(x) = x + 2, we can consider the two compositions f(g(x)) and g(f(x)). In the first, the output of g becomes the input of f.

In the second, the output of f becomes the input of g, If x equals 3, then f(g(x)) = f(5) = 25 and g(f(x)) = g(9) = 11. The order matters in this case. However, you can confirm for yourself that the order does not matter if f(x) = x + 2 and g(x) = x + 4.