Congruence Transformations in Grade 8 - Mathematics - Exercises with solutions

Master congruence transformations: translations, rotations, reflections, and rigid motions through these 10 detailed exercises.

Solutions: Exercises 1 to 10
1 Translation
Exercise 1
Find the coordinates of the image of point A(3, -2) after a translation of 5 units to the right and 4 units up.
Difficulty: Beginner Time: ~2 minutes Skills: Translation Rule
Definition:

Translation: A rigid transformation that moves every point of a figure the same distance in the same direction. Rule: (x, y) → (x+h, y+k) where h is horizontal movement and k is vertical movement.

Note: Translations preserve size, shape, and orientation of figures, moving them without rotation or reflection.

Step-by-step translation method:
  1. Identify the original coordinates (x, y)
  2. Determine the horizontal shift (h) and vertical shift (k)
  3. Add the shifts to the original coordinates: (x+h, y+k)
  4. Write the new coordinates
Original Point
A(3, -2)
Translation Vector
(5, 4)
Translated Point
A'(8, 2)
Step 1: Identify original coordinates

A(3, -2) where x = 3 and y = -2

Step 2: Determine the translation vector

5 units right (h = 5) and 4 units up (k = 4)

Step 3: Apply the translation rule

(x, y) → (x+h, y+k)

(3, -2) → (3+5, -2+4) = (8, 2)

Step 4: Write the translated point

The translated point is A'(8, 2)

The image of A(3, -2) after translation (5, 4) is A'(8, 2)
Final answer:

The coordinates of the image of point A(3, -2) after a translation of 5 units right and 4 units up are A'(8, 2).

Applied rules:

• Translation rule: (x, y) → (x+h, y+k) where (h, k) is the translation vector

• Coordinate transformation: Add translation values to original coordinates

• Verification: Distance between original and translated points equals the magnitude of the translation vector

• Practice Tip: Remember: right/up = positive, left/down = negative

Related Examples:
  • Translation of (2, 5) by vector (-3, 1): (2-3, 5+1) = (-1, 6)
  • Translation of (-4, -1) by vector (6, -3): (-4+6, -1-3) = (2, -4)
  • Translation of (0, 0) by vector (a, b): (0+a, 0+b) = (a, b)
Quick Tips:
  • Positive horizontal values move right, negative move left
  • Positive vertical values move up, negative move down
  • Translations preserve all geometric properties of figures
Frequently Asked Questions:

Q: Does translation change the size or shape of a figure?
A: No, translations are rigid transformations that preserve size, shape, and orientation of figures.

Q: How do I represent a translation as a vector?
A: A translation can be represented as a vector (h, k) where h is horizontal displacement and k is vertical displacement.

2 Rotation
Exercise 2
Find the coordinates of the image of point B(4, 1) after a 90° counterclockwise rotation about the origin.
Difficulty: Beginner Time: ~3 minutes Skills: 90° Rotation Rule
Definition:

Rotation: A rigid transformation that turns a figure around a fixed point called the center of rotation. For 90° counterclockwise rotation about the origin: (x, y) → (-y, x).

Note: Rotations preserve size and shape but change the orientation of figures. The distance from the center of rotation remains constant.

Step-by-step rotation method:
  1. Identify the original coordinates (x, y)
  2. Determine the angle and direction of rotation
  3. Apply the appropriate rotation rule
  4. Write the new coordinates
Original Point
B(4, 1)
Rotation
90° CCW about origin
Rotated Point
B'(-1, 4)
Step 1: Identify original coordinates

B(4, 1) where x = 4 and y = 1

Step 2: Apply 90° counterclockwise rotation rule

Rule: (x, y) → (-y, x)

Step 3: Transform the coordinates

(4, 1) → (-1, 4)

Step 4: Write the rotated point

The rotated point is B'(-1, 4)

The image of B(4, 1) after 90° CCW rotation is B'(-1, 4)
Final answer:

The coordinates of the image of point B(4, 1) after a 90° counterclockwise rotation about the origin are B'(-1, 4).

Applied rules:

• 90° CCW rotation: (x, y) → (-y, x)

• 180° rotation: (x, y) → (-x, -y)

• 270° CCW rotation: (x, y) → (y, -x)

• Practice Tip: Counterclockwise rotations follow the pattern: x→-y→-x→y→x

Related Examples:
  • 90° CCW rotation of (3, 2): (-2, 3)
  • 180° rotation of (-1, 4): (1, -4)
  • 270° CCW rotation of (5, -3): (-3, -5)
Quick Tips:
  • 90° CCW: (x, y) → (-y, x)
  • 180°: (x, y) → (-x, -y)
  • 270° CCW: (x, y) → (y, -x)
Frequently Asked Questions:

Q: What's the difference between clockwise and counterclockwise rotations?
A: Counterclockwise is the positive direction. 90° CW = 270° CCW, and 270° CW = 90° CCW.

Q: Do rotations preserve distances from the center of rotation?
A: Yes, rotations preserve all distances, including the distance from any point to the center of rotation.

3 Reflection
Exercise 3
Find the coordinates of the image of point C(-2, 5) after a reflection over the y-axis.
Difficulty: Beginner Time: ~2 minutes Skills: Y-axis Reflection Rule
Definition:

Reflection: A rigid transformation that creates a mirror image of a figure across a line called the line of reflection. For reflection over y-axis: (x, y) → (-x, y).

Note: Reflections preserve size and shape but reverse orientation. The line of reflection acts as a perpendicular bisector to segments joining corresponding points.

Step-by-step reflection method:
  1. Identify the original coordinates (x, y)
  2. Determine the line of reflection
  3. Apply the appropriate reflection rule
  4. Write the new coordinates
Original Point
C(-2, 5)
Line of Reflection
y-axis
Reflected Point
C'(2, 5)
Step 1: Identify original coordinates

C(-2, 5) where x = -2 and y = 5

Step 2: Apply y-axis reflection rule

Rule: (x, y) → (-x, y)

Step 3: Transform the coordinates

(-2, 5) → (-(-2), 5) = (2, 5)

Step 4: Write the reflected point

The reflected point is C'(2, 5)

The image of C(-2, 5) after reflection over y-axis is C'(2, 5)
Final answer:

The coordinates of the image of point C(-2, 5) after a reflection over the y-axis are C'(2, 5).

Applied rules:

• Y-axis reflection: (x, y) → (-x, y)

• X-axis reflection: (x, y) → (x, -y)

• Y=X reflection: (x, y) → (y, x)

• Practice Tip: Reflection changes the sign of the coordinate perpendicular to the line of reflection

Related Examples:
  • Reflection of (3, 7) over y-axis: (-3, 7)
  • Reflection of (-4, -2) over y-axis: (4, -2)
  • Reflection of (0, 5) over y-axis: (0, 5)
Quick Tips:
  • Y-axis reflection changes the sign of x-coordinate only
  • X-axis reflection changes the sign of y-coordinate only
  • Points on the line of reflection remain unchanged
Frequently Asked Questions:

Q: What happens to points on the line of reflection?
A: Points on the line of reflection remain unchanged after the reflection.

Q: Does reflection preserve orientation?
A: No, reflections reverse orientation, which is why they're sometimes called "improper" isometries.

Solutions: Exercises 4 to 5
4 Identifying transformations
Exercise 4
Triangle ABC has vertices A(1, 2), B(3, 4), C(2, 1). Triangle A'B'C' has vertices A'(4, 2), B'(6, 4), C'(5, 1). What transformation maps triangle ABC to triangle A'B'C'?
Definition:

Identifying transformations: To determine which transformation maps one figure to another, compare corresponding points and look for consistent patterns in coordinate changes. This could be translation, rotation, reflection, or glide reflection.

Note: For congruence transformations, the distance between corresponding points remains constant, and the transformation preserves size and shape.

Step-by-step identification method:
  1. Compare coordinates of corresponding points
  2. Look for consistent changes in x and y values
  3. Determine if the pattern suggests translation, rotation, or reflection
  4. Verify the transformation with all corresponding points
Original Triangle
A(1,2), B(3,4), C(2,1)
Transformed Triangle
A'(4,2), B'(6,4), C'(5,1)
Transformation
Translation (3, 0)
Step 1: Compare corresponding points

A(1, 2) → A'(4, 2)

B(3, 4) → B'(6, 4)

C(2, 1) → C'(5, 1)

Step 2: Look for consistent patterns

A: x increases by 3, y stays same (1→4, 2→2)

B: x increases by 3, y stays same (3→6, 4→4)

C: x increases by 3, y stays same (2→5, 1→1)

Step 3: Identify the transformation

Each point moves 3 units right and 0 units vertically

This is a translation with vector (3, 0)

Step 4: Verify with all points

Rule: (x, y) → (x+3, y)

A(1, 2) → (1+3, 2) = (4, 2) ✓

B(3, 4) → (3+3, 4) = (6, 4) ✓

C(2, 1) → (2+3, 1) = (5, 1) ✓

Triangle A'B'C' is the image of triangle ABC under a translation of 3 units right
Final answer:

The transformation that maps triangle ABC to triangle A'B'C' is a translation of 3 units to the right (translation vector (3, 0)).

Applied rules:

• Translation identification: Look for consistent coordinate changes across all points

• Pattern recognition: Same change in x-coordinates and same change in y-coordinates

• Verification: Apply transformation rule to all corresponding points

• Practice Tip: For translations, all points move by the same vector

Related Examples:
  • If all x-values increase by 2 and y-values decrease by 3: translation vector (2, -3)
  • If (x, y) → (-x, y): reflection over y-axis
  • If (x, y) → (-y, x): 90° counterclockwise rotation about origin
Quick Tips:
  • For translations, look for consistent changes in coordinates
  • For rotations, distances from the center remain constant
  • For reflections, the line of reflection is the perpendicular bisector of segments joining corresponding points
Frequently Asked Questions:

Q: How do I distinguish between different transformations?
A: Translations have consistent coordinate changes, rotations preserve distances from center, reflections create mirror images.

Q: What if the transformation is not obvious?
A: Calculate the distance between corresponding points and look for patterns in how coordinates change.

5 Composite transformations
Exercise 5
Find the coordinates of the image of point D(2, 3) after a translation of (1, -2) followed by a 90° counterclockwise rotation about the origin.
Definition:

Composite transformations: The result of applying two or more transformations sequentially, where the output of one transformation becomes the input for the next. Order matters in composition.

Note: The final result depends on the order of transformations. Perform transformations from right to left (last transformation listed is applied first).

Step-by-step composition method:
  1. Apply the first transformation to the original point
  2. Use the result as input for the second transformation
  3. Continue for all transformations in sequence
  4. Record the final coordinates
Starting Point
D(2, 3)
After Translation
D'(3, 1)
Final Image
D''(-1, 3)
Step 1: Apply first transformation (translation)

Original point: D(2, 3)

Translation vector: (1, -2)

D(2, 3) → D'(2+1, 3-2) = D'(3, 1)

Step 2: Apply second transformation (rotation)

Input: D'(3, 1)

90° CCW rotation rule: (x, y) → (-y, x)

D'(3, 1) → D''(-1, 3)

Step 3: Verify the result

Starting at (2, 3), we ended at (-1, 3)

Step 4: Write the final coordinates

The final coordinates after both transformations are D''(-1, 3)

After translation (1,-2) then 90° CCW rotation, D(2,3) → D''(-1,3)
Final answer:

The coordinates of the image of point D(2, 3) after a translation of (1, -2) followed by a 90° counterclockwise rotation about the origin are D''(-1, 3).

Applied rules:

• Translation: (x, y) → (x+h, y+k)

• 90° CCW rotation: (x, y) → (-y, x)

• Composition order: Apply transformations in the specified sequence

• Practice Tip: Perform transformations step by step to avoid confusion

Related Examples:
  • Point (4, 5) → reflection over x-axis → (4, -5) → translation (2, 1) → (6, -4)
  • Point (-1, 2) → 180° rotation → (1, -2) → reflection over y-axis → (-1, -2)
  • Composition of transformations is generally not commutative
Quick Tips:
  • Perform transformations in the exact order specified
  • Each transformation's output becomes the next transformation's input
  • Composition of transformations is not commutative in general
Frequently Asked Questions:

Q: Does the order of transformations matter?
A: Yes, in general, the order of transformations matters. For example, translating then rotating usually gives a different result than rotating then translating.

Q: What is the result of two reflections over parallel lines?
A: Two reflections over parallel lines result in a translation perpendicular to the lines.

Solutions: Exercises 6 to 10
6 Determining congruence
Exercise 6
Determine if triangle DEF with vertices D(0, 0), E(3, 0), F(0, 4) is congruent to triangle GHI with vertices G(2, 1), H(5, 1), I(2, 5). If so, describe the transformation.
Definition:

Congruent figures: Two figures are congruent if one can be transformed into the other using a sequence of congruence transformations (translations, rotations, reflections). Congruent figures have the same size and shape.

Note: To prove congruence, we must show that corresponding sides are equal in length and that a sequence of transformations exists to map one figure onto the other.

Step-by-step congruence determination method:
  1. Calculate the side lengths of both figures
  2. Compare corresponding side lengths
  3. If sides match, determine the transformation mapping one figure to the other
  4. Verify the transformation maps all corresponding points correctly
Triangle DEF
D(0,0), E(3,0), F(0,4)
Triangle GHI
G(2,1), H(5,1), I(2,5)
Conclusion
Congruent
Step 1: Calculate side lengths of triangle DEF

DE = √[(3-0)² + (0-0)²] = √9 = 3

DF = √[(0-0)² + (4-0)²] = √16 = 4

EF = √[(3-0)² + (0-4)²] = √[9+16] = √25 = 5

Step 2: Calculate side lengths of triangle GHI

GH = √[(5-2)² + (1-1)²] = √9 = 3

GI = √[(2-2)² + (5-1)²] = √16 = 4

HI = √[(5-2)² + (1-5)²] = √[9+16] = √25 = 5

Step 3: Compare corresponding sides

DE = GH = 3

DF = GI = 4

EF = HI = 5

All corresponding sides are equal, so the triangles are congruent

Step 4: Find the transformation mapping DEF to GHI

D(0,0) → G(2,1): requires translation of (2,1)

Check: E(3,0) → (3+2, 0+1) = (5,1) = H ✓

F(0,4) → (0+2, 4+1) = (2,5) = I ✓

The transformation is a translation by vector (2, 1)

Triangle DEF is congruent to triangle GHI via translation (2, 1)
Final answer:

Yes, triangle DEF is congruent to triangle GHI. The corresponding sides are equal (DE = GH = 3, DF = GI = 4, EF = HI = 5), and the transformation that maps triangle DEF to triangle GHI is a translation of 2 units right and 1 unit up (translation vector (2, 1)).

Applied rules:

• Congruence definition: Figures are congruent if they have the same size and shape

• Distance formula: Use to calculate side lengths for comparison

• Transformation identification: Look for consistent mapping between corresponding points

• Practice Tip: For triangles, if all three sides are equal (SSS), the triangles are congruent

Related Examples:
  • Two triangles with sides 3-4-5 are congruent regardless of position/orientation
  • Squares with the same side length are congruent
  • Congruent polygons have equal corresponding sides and angles
Quick Tips:
  • For triangle congruence, check if corresponding sides are equal (SSS criterion)
  • Calculate distances between all corresponding vertices to verify
  • Once congruence is established, find the specific transformation that maps one to the other
Frequently Asked Questions:

Q: What are the criteria for triangle congruence?
A: Common criteria include SSS (side-side-side), SAS (side-angle-side), ASA (angle-side-angle), and AAS (angle-angle-side).

Q: Can congruent figures have different orientations?
A: Yes, congruent figures can be positioned differently as long as they have the same size and shape.

7 Finding transformation rules
Exercise 7
A transformation maps point (x, y) to point (x-3, y+5). Describe this transformation and find the image of point J(7, -2).
Definition:

Transformation rule: A function that describes how each point (x, y) in the pre-image is mapped to a point (x', y') in the image. The rule shows the relationship between original and transformed coordinates.

Note: Transformation rules for translations have the form (x, y) → (x+h, y+k) where (h, k) is the translation vector.

Step-by-step rule analysis method:
  1. Examine the transformation rule to identify the type of transformation
  2. Determine the specific parameters (translation vector, rotation center, etc.)
  3. Apply the rule to the given point
  4. Describe the transformation in geometric terms
Transformation Rule
(x, y) → (x-3, y+5)
Type
Translation
Image of J
J'(4, 3)
Step 1: Analyze the transformation rule

Rule: (x, y) → (x-3, y+5)

This is a translation where x-coordinate decreases by 3 and y-coordinate increases by 5

Step 2: Identify the translation vector

The transformation moves each point 3 units left and 5 units up

Translation vector: (-3, 5)

Step 3: Apply the transformation to point J(7, -2)

J(7, -2) → (7-3, -2+5) = (4, 3)

Step 4: Describe the transformation

This is a translation that moves every point 3 units to the left and 5 units up

Translation by vector (-3, 5); J(7, -2) → J'(4, 3)
Final answer:

The transformation (x, y) → (x-3, y+5) is a translation that moves every point 3 units left and 5 units up (translation vector (-3, 5)). The image of point J(7, -2) is J'(4, 3).

Applied rules:

• Translation rule: (x, y) → (x+h, y+k) represents translation by vector (h, k)

• Vector interpretation: Positive values indicate movement right/up, negative values indicate left/down

• Application: Substitute coordinates into transformation rule

• Practice Tip: Remember: subtracting means moving left/down, adding means moving right/up

Related Examples:
  • Rule (x, y) → (x+4, y-1): Translation vector (4, -1) - 4 right, 1 down
  • Rule (x, y) → (x, -y): Reflection over x-axis
  • Rule (x, y) → (-x, -y): 180° rotation about origin
Quick Tips:
  • Translation rules have form (x, y) → (x+h, y+k)
  • Positive h means right, negative h means left
  • Positive k means up, negative k means down
Frequently Asked Questions:

Q: How do I know if a transformation rule represents a translation?
A: Translation rules add constants to x and y coordinates: (x, y) → (x+h, y+k).

Q: Can transformation rules mix different types of transformations?
A: Yes, rules can represent compositions of transformations like a translation followed by a rotation.

8 Properties of congruence
Exercise 8
Rectangle KLMN has vertices K(1, 1), L(4, 1), M(4, 3), N(1, 3). After a 180° rotation about the origin, find the coordinates of the image rectangle K'L'M'N' and verify that it is congruent to the original rectangle.
Definition:

Properties of congruence transformations: Congruence transformations preserve distances, angles, parallelism, perpendicularity, and the overall shape of figures. The transformed figure is identical to the original except for position and orientation.

Note: All rigid transformations (translations, rotations, reflections) are congruence transformations that preserve geometric properties.

Step-by-step verification method:
  1. Apply the transformation to each vertex of the original figure
  2. Calculate key measurements of both figures
  3. Compare corresponding measurements to verify congruence
  4. Confirm that all geometric properties are preserved
Original Rectangle
K(1,1), L(4,1), M(4,3), N(1,3)
Rotation Rule
180° about origin: (x, y) → (-x, -y)
Image Rectangle
K'(-1,-1), L'(-4,-1), M'(-4,-3), N'(-1,-3)
Step 1: Apply 180° rotation rule to each vertex

Rule: (x, y) → (-x, -y)

K(1, 1) → K'(-1, -1)

L(4, 1) → L'(-4, -1)

M(4, 3) → M'(-4, -3)

N(1, 3) → N'(-1, -3)

Step 2: Verify the shape is preserved

Original: KL = 3, LM = 2, MN = 3, NK = 2 (rectangle)

Image: K'L' = 3, L'M' = 2, M'N' = 3, N'K' = 2 (rectangle)

Step 3: Verify distances are preserved

Original rectangle has length 3 and width 2

Image rectangle has length 3 and width 2

All corresponding sides are equal

Step 4: Confirm congruence

Since all corresponding sides are equal and angles are preserved (still 90°), the rectangles are congruent

Image rectangle K'L'M'N' has vertices K'(-1,-1), L'(-4,-1), M'(-4,-3), N'(-1,-3) and is congruent to KLMN
Final answer:

The coordinates of the image rectangle after a 180° rotation about the origin are K'(-1, -1), L'(-4, -1), M'(-4, -3), and N'(-1, -3). The image rectangle is congruent to the original rectangle because all corresponding sides are equal (length = 3, width = 2) and all angles remain 90°.

Applied rules:

• 180° rotation rule: (x, y) → (-x, -y)

• Congruence preservation: Rigid transformations preserve distances and angles

• Rectangle properties: Opposite sides equal, all angles 90°

• Practice Tip: Rotations about origin follow predictable coordinate changes

Related Examples:
  • Translation preserves all properties: lengths, angles, parallelism, perpendicularity
  • Reflection preserves lengths and angles but reverses orientation
  • Rotation preserves all properties including orientation
Quick Tips:
  • Congruence transformations always preserve distances between points
  • Verify congruence by checking that corresponding sides and angles are equal
  • Parallel lines remain parallel after any rigid transformation
Frequently Asked Questions:

Q: What properties are preserved by all congruence transformations?
A: Distances, angles, parallelism, perpendicularity, collinearity, and area are all preserved.

Q: Do congruence transformations change the perimeter of figures?
A: No, since distances are preserved, the perimeter remains unchanged after any congruence transformation.

9 Real-world application
Exercise 9
A company logo consists of a triangle with vertices at (2, 3), (6, 3), and (4, 7). For a new design, they rotate the logo 90° counterclockwise about the origin and then translate it 5 units right and 2 units down. Find the new coordinates of the logo.
Definition:

Real-world applications of transformations: Congruence transformations model physical movements, design patterns, manufacturing processes, and computer graphics. The properties of congruence ensure that the transformed object maintains its essential characteristics.

Note: In practical applications, transformations might be composed to achieve desired positioning and orientation of objects.

Step-by-step application method:
  1. Apply the first transformation (rotation) to all vertices
  2. Apply the second transformation (translation) to the results
  3. Record the final coordinates after both transformations
  4. Verify that the final figure is congruent to the original
Original Logo
(2,3), (6,3), (4,7)
After Rotation
(-3,2), (-3,6), (-7,4)
Final Position
(2,-0), (2,4), (-2,2)
Step 1: Apply 90° counterclockwise rotation to each vertex

Rule: (x, y) → (-y, x)

(2, 3) → (-3, 2)

(6, 3) → (-3, 6)

(4, 7) → (-7, 4)

Step 2: Apply translation of 5 units right and 2 units down

Rule: (x, y) → (x+5, y-2)

(-3, 2) → (-3+5, 2-2) = (2, 0)

(-3, 6) → (-3+5, 6-2) = (2, 4)

(-7, 4) → (-7+5, 4-2) = (-2, 2)

Step 3: Record final coordinates

The new logo has vertices at (2, 0), (2, 4), and (-2, 2)

Step 4: Verify congruence

Both transformations are rigid, so the final triangle is congruent to the original

New logo vertices: (2, 0), (2, 4), (-2, 2)
Final answer:

After rotating the logo 90° counterclockwise about the origin and then translating it 5 units right and 2 units down, the new coordinates of the logo are (2, 0), (2, 4), and (-2, 2). The final logo is congruent to the original since both transformations are rigid motions.

Applied rules:

• 90° CCW rotation: (x, y) → (-y, x)

• Translation: (x, y) → (x+h, y+k)

• Composite transformations: Apply in sequence from first to last

• Practice Tip: Real-world designs often use transformations for symmetry and positioning

Related Examples:
  • Manufacturing: Parts moved by robotic arms using translations and rotations
  • Architecture: Building designs mirrored using reflections
  • Computer graphics: Objects repositioned using transformation matrices
Quick Tips:
  • Real-world transformations often involve multiple steps
  • Always perform transformations in the specified order
  • Congruence ensures the transformed object maintains its functional properties
Frequently Asked Questions:

Q: Why are congruence transformations important in design?
A: They ensure that transformed objects maintain the same functional properties as the original, which is crucial for engineering and architectural applications.

Q: Can transformations be reversed?
A: Yes, all congruence transformations have inverses that return the figure to its original position.

10 Advanced problem solving
Exercise 10
Triangle PQR has vertices P(a, b), Q(c, d), and R(e, f). After a transformation, the image triangle P'Q'R' has vertices P'(b, -a), Q'(d, -c), and R'(f, -e). Identify the transformation and prove it preserves congruence.
Definition:

General transformation analysis: When working with variables, identify the transformation by examining the pattern of coordinate changes. Prove congruence by showing that distances between points remain unchanged.

Note: The transformation (x, y) → (y, -x) represents a 270° counterclockwise rotation (or 90° clockwise rotation) about the origin.

Step-by-step analysis method:
  1. Identify the transformation rule by comparing input and output coordinates
  2. Recognize the type of transformation based on the rule
  3. Prove congruence by showing distance preservation using the distance formula
  4. Verify with specific examples if possible
Original Triangle
P(a,b), Q(c,d), R(e,f)
Transformation Rule
(x, y) → (y, -x)
Transformation Type
270° CCW rotation
Step 1: Identify the transformation rule

Compare original and image points:

P(a, b) → P'(b, -a)

Q(c, d) → Q'(d, -c)

R(e, f) → R'(f, -e)

The pattern is: (x, y) → (y, -x)

Step 2: Identify the transformation type

The rule (x, y) → (y, -x) represents a 270° counterclockwise rotation about the origin (or equivalently, a 90° clockwise rotation).

Step 3: Prove distance preservation

Distance between P(a, b) and Q(c, d):

d₁ = √[(c-a)² + (d-b)²]

Distance between P'(b, -a) and Q'(d, -c):

d₂ = √[(d-b)² + (-c-(-a))²] = √[(d-b)² + (a-c)²] = √[(d-b)² + (c-a)²]

Therefore, d₁ = d₂, so distances are preserved.

Step 4: General proof of congruence

Since the transformation (x, y) → (y, -x) is a rotation, it's a rigid transformation.

All rotations preserve distances, angles, and overall shape.

Therefore, triangle P'Q'R' is congruent to triangle PQR.

Transformation: (x, y) → (y, -x) = 270° CCW rotation; Congruence preserved
Final answer:

The transformation is (x, y) → (y, -x), which represents a 270° counterclockwise rotation (or 90° clockwise rotation) about the origin. This transformation preserves congruence because it's a rigid motion that maintains all distances between points. For any two points (x₁, y₁) and (x₂, y₂), the distance between them equals the distance between their images (y₁, -x₁) and (y₂, -x₂).

Applied rules:

• 270° CCW rotation: (x, y) → (y, -x)

• Distance preservation: Use distance formula to prove preservation

• Rigid transformation property: Rotations preserve all geometric properties

• Practice Tip: When working with variables, focus on the transformation rule pattern

Related Examples:
  • (x, y) → (-y, x): 90° CCW rotation
  • (x, y) → (-x, -y): 180° rotation
  • (x, y) → (y, x): Reflection over line y=x
Quick Tips:
  • Look for coordinate patterns when working with variables
  • Prove congruence by showing distance preservation
  • Rotations, translations, and reflections are all rigid transformations
Frequently Asked Questions:

Q: How do I prove a transformation preserves congruence?
A: Show that the distance between any two points equals the distance between their images using the distance formula.

Q: What's the difference between 270° CCW and 90° CW?
A: They are the same rotation! A 270° counterclockwise rotation is equivalent to a 90° clockwise rotation.

Key Laws, Methods, Rules, and Definitions
\((x, y) \rightarrow (x+h, y+k)\)
Translation Rule
Key definitions:

Congruence Transformation: A transformation that preserves the size and shape of a figure. Also called an isometry or rigid transformation.

Isometry: A transformation that preserves distances between points.

Congruent Figures: Figures that have the same size and shape; one can be transformed into the other using congruence transformations.

Complete methodology:
  1. Identify the transformation type: Determine if it's translation, rotation, reflection, or glide reflection
  2. Apply the appropriate rule: Use the specific transformation rule for coordinates
  3. Verify the result: Check that distances and angles are preserved
  4. Confirm congruence: Ensure the transformed figure is congruent to the original
Tip 1: Translations: (x, y) → (x+h, y+k) where (h, k) is the translation vector.
Tip 2: Rotations about origin: 90°CCW: (x, y) → (-y, x), 180°: (x, y) → (-x, -y).
Tip 3: Reflections: x-axis: (x, y) → (x, -y), y-axis: (x, y) → (-x, y).
Tip 4: Congruence transformations preserve distances, angles, parallelism, and orientation (except for reflections).
Common errors: Confusing transformation rules, forgetting that order matters in compositions, misapplying rotation directions.
Exam preparation: Memorize transformation rules, practice composite transformations, understand congruence criteria.
Formulas to memorize:

• Translation: \((x, y) \rightarrow (x+h, y+k)\)

• 90° CCW rotation: \((x, y) \rightarrow (-y, x)\)

• 180° rotation: \((x, y) \rightarrow (-x, -y)\)

• 270° CCW rotation: \((x, y) \rightarrow (y, -x)\)

• X-axis reflection: \((x, y) \rightarrow (x, -y)\)

• Y-axis reflection: \((x, y) \rightarrow (-x, y)\)

Rules and Methods for Congruence Transformations
\((x, y) \rightarrow (-y, x)\)
90° CCW Rotation
Translation
\((x, y) \rightarrow (x+h, y+k)\)
Slide in direction (h, k)
90° Rotation
\((x, y) \rightarrow (-y, x)\)
Counterclockwise about origin
Reflection
\((x, y) \rightarrow (-x, y)\)
Over y-axis

Key Takeaways

  • Congruence transformations preserve size, shape, and all geometric properties
  • Translations, rotations, and reflections are all rigid transformations
  • Composite transformations are performed in sequence from first to last
  • Order of transformations matters in compositions
  • Congruent figures have equal corresponding sides and angles

Questions & Answers

Question: I'm having trouble distinguishing between the different rotation rules. Is there a pattern I can remember?

Answer: Yes, there's a clear pattern for counterclockwise rotations about the origin:

  • 90° CCW: (x, y) → (-y, x) - y goes to x-position with negative sign
  • 180° CCW: (x, y) → (-x, -y) - both coordinates change signs
  • 270° CCW: (x, y) → (y, -x) - x goes to y-position with negative sign

You can remember the 90° rule as "flip and negate the first": switch x and y, then negate what was originally x. For 270°, it's similar but you negate what was originally y.

Question: How can I tell if two figures are congruent without measuring all their sides?

Answer: You can determine congruence by finding a sequence of rigid transformations that maps one figure onto the other:

  • Check for transformation possibility: If you can translate, rotate, or reflect one figure to perfectly overlap the other, they are congruent
  • Compare key points: If corresponding vertices have the same relative positions after a transformation, the figures are congruent
  • Use congruence criteria: For triangles, SSS, SAS, ASA, or AAS criteria can establish congruence

If a rigid transformation exists that maps one figure exactly onto the other, they are congruent by definition.

Question: Does the order matter when applying multiple transformations?

Answer: Yes, the order of transformations generally matters. For example:

  • If you translate a point and then rotate it, you'll get a different result than if you rotate it first and then translate it
  • Some special cases commute: two translations always commute, and two rotations about the same center commute
  • However, rotation followed by translation typically gives a different result than translation followed by rotation

Always follow the order specified in the problem: apply transformations from first mentioned to last mentioned.

Geometry Glossary

Congruence Transformation
A transformation that preserves the size and shape of a figure. Also called an isometry or rigid transformation. Includes translations, rotations, and reflections.
Isometry
A transformation that preserves distances between points. All congruence transformations are isometries.
Translation
A rigid transformation that moves every point of a figure the same distance in the same direction. Rule: (x, y) → (x+h, y+k).
Rotation
A rigid transformation that turns a figure around a fixed point (center of rotation) by a specified angle and direction.
Reflection
A rigid transformation that creates a mirror image of a figure across a line (line of reflection).
Congruent Figures
Figures that have the same size and shape. One can be transformed into the other using a sequence of congruence transformations.
Rigid Motion
Another term for congruence transformation. A transformation that preserves distances and angles.
Composite Transformation
The result of applying two or more transformations in sequence. The output of one transformation becomes the input for the next.

Congruence Transformations Educational Team

Certified Mathematics Educators & Curriculum Specialists

Our team of experienced middle school math teachers and geometry specialists creates research-based, student-friendly resources focused on geometric transformations and congruence. All content is aligned with Common Core State Standards and reviewed by mathematics education experts to ensure accuracy and pedagogical effectiveness.