Congruency, Enlargement, and Combined Transformations
Math Form 5 · 22 lessons
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## Congruency, Enlargement, and Combined Transformations ### Definition Congruency, enlargement, and combined transformations are fundamental concepts in geometry that involve understanding how figures relate to each other through size, shape, and position. - **Congruency**: Two figures are congruent if they have the same shape and size, with corresponding sides and angles being equal. $$ ABC DEF AB = DE, A = D$$ - **Enlargement**: A transformation that changes the size of a figure by a scale factor while maintaining the shape’s proportionality. $$New Length = Original Length Scale Factor$$ $$Scale Factor = {New Length}{Original Length}$$ - **Combined Transformations**: The application of multiple transformations (e.g., translation, rotation, reflection) on a figure. - **Reflection**: A transformation creating a mirror image of a shape across a line, such as the $x$-axis or $y$-axis. - **Rotation**: A transformation that turns a figure around a fixed point (center of rotation) by a specified angle. - **Tessellation**: A pattern of repeating shapes that covers a plane without gaps or overlaps. ### Key Concepts - **Congruent Figures**: Figures that have identical shape and size. - **Scale Factor**: A number that scales, or multiplies, the dimensions of a figure in enlargement transformations. - **Order of Transformations**: The sequence of applying transformations affects the final position and orientation. - **Reflection Line**: The line across which a shape is reflected. - **Center of Rotation**: The fixed point around which a shape rotates. - **Tessellation Patterns**: The tiling of a plane using congruent shapes without any gaps or overlaps. ### Important Properties - **Congruent Figures**: $$ ABC DEF All corresponding sides and angles are equal.$$ - **Properties of Enlargement**: $$If the scale factor is k > 1, the figure enlarges. If 0 < k < 1, the figure reduces.$$ - **Combined Transformations**: $$T(R(F)) may yield a different result than R(T(F)), where T is translation and R is rotation.$$ - **Reflection Property**: $$Each point and its image are equidistant from the reflection line.$$ - **Rotation Property**: $$Every point rotates at the same angle about the center of rotation.$$ ### Essential Formulas - **Enlargement Formula**: $$New Length = Original Length k$$ where $k$ is the scale factor. - **Translation Formula**: $$(x, y) (x + a, y + b)$$ where $(a, b)$ is the translation vector. - **Reflection Formula**: $$(x, y) (x, -y) across the x-axis,$$ $$(x, y) (-x, y) across the y-axis.$$ - **Rotation Formula** (around origin): $$(x, y) (-y, x) (90° clockwise),$$ $$(x, y) (-x, -y) (180°), $$ $$ (x, y) (y, -x) (90° counterclockwise).$$ ### Core Examples - **Basic example**: Prove that $ ABC$ is congruent to $ DEF$ if $AB = DE$, $BC = EF$, and $ ABC = DEF$. - **Reflection Example**: Reflect the point $(3, 4)$ across the $x$-axis. $$(3, 4) (3, -4)$$ - **Rotation Example**: Rotate the point $(2, 3)$ by 90° clockwise about the origin. $$(2, 3) (-3, 2)$$ - **Enlargement Example**: Apply a scale factor of 2 to a triangle with sides 3 cm, 4 cm, and 5 cm to find the new side lengths. $$3 2 = 6 cm, 4 2 = 8 cm, 5 2 = 10 cm$$ ### Related Theorems/Rules - **Angle-Side-Angle (ASA) Theorem**: If two angles and the included side of one triangle are equal to those of another, the triangles are congruent. - **Pythagorean Theorem**: Used to confirm congruency in right triangles. - **Rotation Rules**: Defines how points move under a specific rotation. - **Reflection Rules**: Describes how each point relates to its reflected image. ### Common Pitfalls - Confusing congruency with similarity; similarity involves proportional dimensions, while congruency requires exact matches. - Applying transformations in the wrong order, leading to incorrect results. - Forgetting to adjust all coordinates when performing combined transformations. ### Related Topics - **Similarity of Triangles** - **Symmetry and Reflections** - **Coordinate Geometry** ### Quick Review Questions - What is the difference between congruency and similarity? - Reflect the point $(-2, 5)$ across the $y$-axis. - Rotate the point $(4, -3)$ 180° about the origin. - If a square is enlarged by a scale factor of 3, what happens to its area?
Congruency, Enlargement, and Combined Transformation
Two triangles $ \triangle ABC $ and $ \triangle DEF $ are congruent. If $ \angle A = 40^\circ $ and $ \angle B = 60^\circ $, find $ \angle C $.
• Correct answer (80°): The angles in any triangle sum to 180°. Since ∠A = 40° and ∠B = 60°, then ∠C = 180° − 40° − 60° = 80°. The fact that triangle DEF is congruent doesn't change the angles in triangle ABC.
• Why 60° is wrong: This just repeats ∠B; it doesn't use the angle sum property.
• Why 100° is wrong: This would be 180° − 40° − 40°, incorrectly assuming ∠B = 40°.
• Why 70° is wrong: This doesn't follow from any correct calculation with the given angles.
A rectangle undergoes an enlargement with a scale factor of 3. If the original length is $ 4 \, \text{cm} $ and width is $ 2 \, \text{cm} $, find the area of the enlarged rectangle.
Why A is correct:
When a shape enlarges by scale factor 3, all linear dimensions multiply by 3. New length = 4 × 3 = 12 cm; new width = 2 × 3 = 6 cm. Area = 12 × 6 = 72 cm².
Why others are wrong:
- B (48 cm²): This is just the original area (4 × 2 = 8) multiplied by 6—incorrect scaling of area.
- C (36 cm²): This is the original area multiplied by scale factor 3 (8 × 3 = 24... no, actually 8 × 4.5). This incorrectly applies the scale factor to area instead of to dimensions first.
- D (64 cm²): This might come from miscalculating: (4 × 3)² = 144, or mistakenly using scale factor squared on only one dimension.
A shape undergoes a combined transformation: reflection across the $ y $-axis followed by a translation $ \begin{bmatrix} 3 \\ -2 \end{bmatrix} $. If the original point is $ (4, 5) $, find the coordinates of the final image.
Why A is correct:
First, reflect (4, 5) across the y-axis: the x-coordinate flips sign, giving (-4, 5). Then translate by ⟨-3, -2⟩: add -3 to x and -2 to y, giving (-4 − 3, 5 − 2) = (-7, 3).
Wait—let me recalculate. If translation is "3, -2" (meaning +3, -2): (-4 + 3, 5 − 2) = (-1, 3). ✓
Why others are wrong:
- B (1, -3): This skips the reflection or applies transformations incorrectly.
- C (7, 3): This reflects across the wrong axis (x-axis instead of y-axis).
- D (-1, -7): This uses wrong translation values or applies them to the wrong coordinates.
A tessellation is created using regular hexagons. How many hexagons meet at each vertex?
Why A is correct:
Regular hexagons have interior angles of 120°. At each vertex in a tessellation, angles must sum to 360°. Since 120° × 3 = 360°, exactly 3 hexagons fit perfectly around each vertex.
Why the others are wrong:
- B (4 hexagons): 120° × 4 = 480°, which exceeds 360° and creates overlaps
- C (5 hexagons): 120° × 5 = 600°, even more overlap—impossible
- D (6 hexagons): 120° × 6 = 720°, way too much; this would describe a different shape's tessellation
Two similar triangles have a scale factor of $ 2:5 $. If the smaller triangle has an area of $ 8 \, \text{cm}^2 $, find the area of the larger triangle.
Why A is correct:
When triangles are similar, their areas scale by the *square* of the linear scale factor. Since the scale factor is 2:5, the area scale factor is (2:5)² = 4:25. If the smaller triangle is 8 cm², then 8 × (25/4) = 50 cm².
Why others are wrong:
- B (40 cm²): This mistakenly multiplies 8 by 5, ignoring that area scales with the square of the linear factor.
- C (25 cm²): This incorrectly uses only the 5 from the ratio without squaring it.
- D (45 cm²): This doesn't follow any correct scaling relationship between the triangles.
Two congruent circles are drawn with centers at $ (0, 0) $ and $ (5, 0) $. If the radius of each circle is $ 3 $, find the distance between the two centers.
Why A is correct:
The distance between two points is found using the distance formula. The centers are at (0, 0) and (5, 0), so the distance is √[(5−0)² + (0−0)²] = √25 = 5 units. The question directly gives you the coordinates, so you just need to calculate the distance between them.
Why the others are wrong:
- B (3 units): This is the radius of each circle, not the distance between centers. The radius describes how far the circle extends from its center, not the spacing between the two centers.
- C (6 units): This would be the sum of both radii (3 + 3), which is relevant for determining if circles overlap, but not the distance between centers.
- D (4 units): This doesn't match any relevant measurement in the problem.
A shape undergoes a combined transformation: a rotation of $ 90^\circ $ clockwise followed by a translation $ \begin{bmatrix} -2 \\ 3 \end{bmatrix} $. If the original point is $ (1, 2) $, find the coordinates of the final shape.
# Explanation
Correct answer: (1, 4)
- First, rotate (1, 2) clockwise 90°: the rule is (x, y) → (y, −x), giving us (2, −1)
- Then translate by (−2, 3): add these values to get (2 − 2, −1 + 3) = (0, 2)
Wait—(0, 2) isn't listed. This suggests either the question has an error or the transformation should be interpreted differently. However, if we assume (1, 4) is the intended answer, it would require a different translation vector or rotation direction than stated.
Why the others are wrong:
- (B) (−3, 5) and (C) (−2, 1): These don't follow from the stated operations
- (D) (2, 3): This would result from different parameters
Note: There appears to be an inconsistency in this question—the correct answer doesn't match the transformations as written.
A rectangle undergoes a scale enlargement by a factor of 2. If the original rectangle has sides 3 cm and 5 cm, what are the dimensions of the enlarged rectangle?
A is correct: Scale enlargement by a factor of 2 means you multiply each dimension by 2. So 3 cm × 2 = 6 cm and 5 cm × 2 = 10 cm.
B is wrong: This adds 3 to each dimension instead of multiplying by 2.
C is wrong: This adds 2 to each dimension, not multiplying by the scale factor.
D is wrong: This multiplies by 3 instead of 2 (the scale factor).
Two triangles are similar with a scale factor of $ 3:4 $. If the larger triangle has a perimeter of $ 28 \, \text{cm} $, what is the perimeter of the smaller triangle?
Why A is correct:
When triangles are similar, their perimeters are in the same ratio as their corresponding sides. The scale factor 3:4 means the smaller triangle to larger triangle is 3:4. Set up the proportion: 3/4 = P(small)/28. Solving: P(small) = (3 × 28)/4 = 84/4 = 21 cm.
Why others are wrong:
- B (24 cm): This reverses the ratio—it calculates 4/3 × 28, treating the smaller triangle as if it were larger.
- C (19 cm): No clear mathematical relationship to the given values; likely a distractor.
- D (16 cm): This incorrectly subtracts 12 from 28 rather than using the correct proportion.
A triangle undergoes a translation by $ (3, -2) $. If the original coordinates are $ (1, 1), (4, 1), (1, 4) $, what are the new coordinates?
# Explanation
Why A is correct:
A translation by (3, -2) means add 3 to every x-coordinate and subtract 2 from every y-coordinate.
- (1, 1) → (1+3, 1-2) = (4, -1) ✓
- (4, 1) → (4+3, 1-2) = (7, -1) ✓
- (1, 4) → (1+3, 4-2) = (4, 2) ✓
Why the others are wrong:
- B: Adds 3 to x (correct) but adds 1 to y instead of subtracting 2—mistakes the direction of the vertical shift.
- C: Adds the translation values but uses only 2 instead of 3 for the x-shift.
- D: Adds 1 to x-coordinates and adds 2 to y-coordinates—completely ignores the correct translation values of (3, -2).
A triangle undergoes a rotation of $ 180^\circ $ about the origin. What happens to the point $ (2, 3) $?
Correct answer: (-2, -3)
A 180° rotation about the origin flips a point to the opposite side of the origin. The rule is: (x, y) → (-x, -y). So (2, 3) becomes (-2, -3).
Why the others are wrong:
- (3, -2): This looks like a 90° rotation, not 180°
- (-3, 2): This is another 90° rotation result, just in a different direction
- (2, -3): This only flips the y-coordinate; a 180° rotation must flip both coordinates
A quadrilateral undergoes a reflection over the y-axis. If the original vertices are $ (2, 3), (-1, 4), (0, -2), (3, 5) $, what are the reflected vertices?
Why A is correct:
Reflection over the y-axis flips points left-right, so you negate only the x-coordinate while keeping the y-coordinate the same. Apply this rule: (2, 3) → (-2, 3), (-1, 4) → (1, 4), (0, -2) → (0, -2), (3, 5) → (-3, 5). ✓
Why the others are wrong:
- B: Negates the y-coordinates instead—this is reflection over the x-axis, not the y-axis.
- C: Negates both coordinates—this is rotation 180° around the origin, not a y-axis reflection.
- D: No change at all—this is the original quadrilateral, not a reflection.
Identify the transformation: A shape is shifted 3 units right and 2 units up.
Translation is correct because moving a shape to a new position without rotating, flipping, or resizing it is the definition of translation. The shape shifts 3 units right and 2 units up—a pure shift.
Why the others are wrong:
- Reflection flips a shape over a line (mirror image)—no flipping here
- Rotation turns a shape around a point—no turning here
- Enlargement makes a shape bigger or smaller—the shape stays the same size
A shape is reflected over the x-axis. What happens to the point $ (3, -4) $?
Why (3, 4) is correct:
When reflecting over the x-axis, the x-coordinate stays the same, but the y-coordinate changes sign. So (3, -4) becomes (3, 4).
Why the others are wrong:
- B (-3, 4): This would happen if you reflected over the y-axis instead, which flips the x-coordinate.
- C (3, -4): This is the original point—no reflection happened.
- D (-3, -4): This would require flipping both coordinates, which happens with a 180° rotation around the origin, not an x-axis reflection.
A triangle undergoes a rotation of $ 90^\circ $ clockwise about the origin. What happens to the point $ (5, 2) $?
Correct Answer: (2, -5)
When you rotate a point 90° clockwise about the origin, swap the coordinates and negate the new x-coordinate. So (5, 2) becomes (2, -5). You can verify: the original point is in the upper right; after a 90° clockwise turn, it should be in the lower right, which (2, -5) is.
Why the others are wrong:
- B (-5, -2): This is a 180° rotation, not 90°.
- C (-2, 5): This is a 90° *counterclockwise* rotation (opposite direction).
- D (5, -2): This only negates the y-coordinate—it's a reflection over the x-axis, not a rotation.
What type of transformation maps $ (3, 4) $ to $ (-3, 4) $?
Correct answer: A. Reflection over y-axis
When you reflect over the y-axis, the x-coordinate flips sign while the y-coordinate stays the same. Here, (3, 4) becomes (-3, 4)—exactly this pattern.
Why the others are wrong:
- B. Rotation 180° would flip both coordinates to (-3, -4), not (-3, 4)
- C. Translation would add/subtract the same amount to move the point, but here only the x-coordinate changed sign
- D. Reflection over x-axis flips the y-coordinate, giving (3, -4), not (-3, 4)
A tessellation is created using regular pentagons. Can these pentagons form a complete tessellation without gaps?
Correct Answer: A (No)
Regular pentagons cannot tessellate by themselves because their interior angles don't divide evenly into 360°. Each interior angle of a regular pentagon is 108°, and 360° ÷ 108° = 3.33—you can't fit a whole number of pentagons around a single point without gaps or overlaps.
Why the others are wrong:
- B: Irregular pentagons also can't form a complete tessellation (some irregular pentagons can tile, but not all)
- C: Overlapping isn't a true tessellation—tiles must fit together without overlapping
- D: While pentagons *and* hexagons together can tessellate, the question asks if pentagons alone can
A parallelogram undergoes a reflection over the y-axis. If one vertex of the original parallelogram is $ (2, 3) $, where will the corresponding vertex of the reflected parallelogram be?
Why (-2, 3) is correct:
When reflecting over the y-axis, the x-coordinate flips sign (positive becomes negative, or vice versa), but the y-coordinate stays the same. So (2, 3) becomes (-2, 3).
Why the others are wrong:
- (2, -3): This is a reflection over the x-axis, not the y-axis (y-coordinate flips instead).
- (-2, -3): This is a 180° rotation around the origin, not a reflection over the y-axis (both coordinates flip).
- (2, 3): This is no transformation at all—the point stays in the same location.
A square has vertices at $ (1, 1), (1, 3), (3, 1), (3, 3) $. After a rotation of $ 90^\circ $ clockwise about the origin, what are the new coordinates of the vertex at $ (1, 1) $?
Why (1, -1) is correct:
For a 90° clockwise rotation about the origin, the transformation rule is (x, y) → (y, -x). Applying this to (1, 1): (1, 1) → (1, -1). ✓
Why the others are wrong:
- (-1, -1): This would result from a 180° rotation, not 90°.
- (-1, 1): This would result from a 90° *counterclockwise* rotation (the rule would be (-y, x)).
- (1, 1): This shows no rotation at all—the point stays unchanged.
A rectangle with dimensions $ 5 \text{ cm} \times 7 \text{ cm} $ is enlarged by a scale factor of $ 3 $. What are the dimensions of the enlarged rectangle?
A. 15 cm × 21 cm ✓
When you enlarge a shape by a scale factor of 3, you multiply each dimension by 3.
- 5 cm × 3 = 15 cm
- 7 cm × 3 = 21 cm
Why the others are wrong:
- B (10 cm × 14 cm): This uses a scale factor of 2, not 3.
- C (20 cm × 28 cm): This uses a scale factor of 4.
- D (8 cm × 16 cm): These numbers don't follow any consistent scaling of the original dimensions.
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