Similarity Transformations in Grade 8 - Mathematics - Exercises with solutions

Master similarity transformations: dilations, scale factors, proportional reasoning, and geometric similarity through these 10 detailed exercises.

Solutions: Exercises 1 to 10
1 Dilation with scale factor
Exercise 1
Find the coordinates of the image of point A(2, 4) after a dilation centered at the origin with a scale factor of 3.
Difficulty: Beginner Time: ~2 minutes Skills: Dilation Rule
Definition:

Dilation: A similarity transformation that enlarges or reduces a figure by a scale factor from a center point. Rule: (x, y) → (kx, ky) where k is the scale factor and center is at origin.

Note: If k > 1, the figure is enlarged; if 0 < k < 1, the figure is reduced; if k < 0, the figure is reflected and scaled.

Step-by-step dilation method:
  1. Identify the original coordinates (x, y)
  2. Determine the scale factor (k)
  3. Multiply both coordinates by the scale factor
  4. Write the new coordinates (kx, ky)
Original Point
A(2, 4)
Scale Factor
k = 3
Dilated Point
A'(6, 12)
Step 1: Identify original coordinates

A(2, 4) where x = 2 and y = 4

Step 2: Identify the scale factor

k = 3 (figure will be enlarged by a factor of 3)

Step 3: Apply the dilation rule

(x, y) → (kx, ky)

(2, 4) → (3×2, 3×4) = (6, 12)

Step 4: Write the dilated point

The dilated point is A'(6, 12)

The image of A(2, 4) after dilation with scale factor 3 is A'(6, 12)
Final answer:

The coordinates of the image of point A(2, 4) after a dilation centered at the origin with a scale factor of 3 are A'(6, 12).

Applied rules:

• Dilation rule (origin center): (x, y) → (kx, ky)

• Scale factor effect: k > 1 enlarges, 0 < k < 1 reduces

• Coordinate transformation: Multiply each coordinate by scale factor

• Practice Tip: Remember: scale factor multiplies both x and y coordinates equally

Related Examples:
  • Dilation of (3, 5) with scale factor 2: (6, 10)
  • Dilation of (4, 8) with scale factor 0.5: (2, 4)
  • Dilation of (-2, 3) with scale factor -2: (4, -6)
Quick Tips:
  • Positive scale factors preserve orientation
  • Negative scale factors reflect and scale the figure
  • Scale factor of 1 leaves the figure unchanged
Frequently Asked Questions:

Q: What happens when the scale factor is between 0 and 1?
A: The figure is reduced in size while maintaining the same shape and orientation.

Q: How does dilation affect the area of a figure?
A: The area changes by the square of the scale factor (area_new = k² × area_original).

2 Dilation about origin
Exercise 2
Triangle ABC has vertices A(1, 2), B(3, 1), and C(2, 4). Find the coordinates of the image triangle A'B'C' after a dilation centered at the origin with a scale factor of 2.
Difficulty: Beginner Time: ~3 minutes Skills: Dilation of Polygons
Definition:

Dilation of a polygon: To dilate a polygon, apply the dilation rule to each vertex individually. For a dilation centered at the origin with scale factor k: (x, y) → (kx, ky).

Note: The dilated polygon is similar to the original polygon, meaning corresponding angles are equal and corresponding sides are proportional.

Step-by-step polygon dilation method:
  1. Identify the coordinates of each vertex of the original polygon
  2. Apply the dilation rule to each vertex: (x, y) → (kx, ky)
  3. Record the coordinates of the dilated vertices
  4. Connect the dilated vertices to form the dilated polygon
Original Triangle
A(1,2), B(3,1), C(2,4)
Scale Factor
k = 2
Dilated Triangle
A'(2,4), B'(6,2), C'(4,8)
Step 1: Identify original coordinates

A(1, 2), B(3, 1), C(2, 4)

Step 2: Apply dilation rule to each vertex

Rule: (x, y) → (2x, 2y)

A(1, 2) → A'(2×1, 2×2) = A'(2, 4)

B(3, 1) → B'(2×3, 2×1) = B'(6, 2)

C(2, 4) → C'(2×2, 2×4) = C'(4, 8)

Step 3: Verify the dilation

Each coordinate was multiplied by the scale factor k = 2

Step 4: Write the dilated triangle coordinates

Triangle A'B'C' has vertices A'(2, 4), B'(6, 2), and C'(4, 8)

Triangle A'B'C' has vertices A'(2,4), B'(6,2), C'(4,8)
Final answer:

The coordinates of the image triangle A'B'C' after a dilation centered at the origin with a scale factor of 2 are A'(2, 4), B'(6, 2), and C'(4, 8).

Applied rules:

• Dilation rule: (x, y) → (kx, ky) applied to each vertex

• Vertex transformation: Apply rule to each vertex independently

• Similarity preservation: Dilated triangle is similar to original

• Practice Tip: Always dilate each vertex separately before connecting them

Related Examples:
  • Triangle with vertices (1,1), (3,1), (2,4) dilated by factor 3: (3,3), (9,3), (6,12)
  • Square with vertices (0,0), (2,0), (2,2), (0,2) dilated by factor 0.5: (0,0), (1,0), (1,1), (0,1)
  • Rectangle dilation preserves all angle measures
Quick Tips:
  • Dilate each vertex separately before connecting them
  • The dilated figure is similar to the original figure
  • Corresponding angles remain equal after dilation
Frequently Asked Questions:

Q: Does dilation change the angles of a figure?
A: No, dilation preserves angle measures. All corresponding angles remain equal.

Q: How do I know if two figures are similar?
A: Two figures are similar if corresponding angles are equal and corresponding sides are proportional.

3 Finding scale factor
Exercise 3
Point P(3, 6) is dilated to point P'(9, 18) with the center of dilation at the origin. Find the scale factor of the dilation.
Difficulty: Intermediate Time: ~3 minutes Skills: Scale Factor Calculation
Definition:

Scale factor: The ratio of the distance from the center of dilation to an image point compared to the distance from the center to the pre-image point. k = distance to image / distance to pre-image.

Note: For a dilation centered at the origin, the scale factor can be found by dividing any coordinate of the image by the corresponding coordinate of the pre-image: k = x'/x = y'/y.

Step-by-step scale factor method:
  1. Identify the coordinates of the pre-image point (x, y)
  2. Identify the coordinates of the image point (x', y')
  3. Calculate the scale factor using k = x'/x or k = y'/y
  4. Verify with the other coordinate to ensure consistency
Pre-image Point
P(3, 6)
Image Point
P'(9, 18)
Scale Factor
k = 3
Step 1: Identify the coordinates

Pre-image: P(3, 6), Image: P'(9, 18)

Step 2: Calculate scale factor using x-coordinates

k = x'/x = 9/3 = 3

Step 3: Verify using y-coordinates

k = y'/y = 18/6 = 3

Both calculations give the same result

Step 4: Confirm the scale factor

The scale factor is k = 3

The scale factor of the dilation is k = 3
Final answer:

The scale factor of the dilation is k = 3.

Applied rules:

• Scale factor formula: k = image coordinate / pre-image coordinate

• Consistency check: Use both x and y coordinates to verify

• Origin center property: For origin center, k = x'/x = y'/y

• Practice Tip: Always verify with both coordinates to ensure accuracy

Related Examples:
  • If (2, 4) → (6, 12), then k = 6/2 = 3 or 12/4 = 3
  • If (5, 10) → (2.5, 5), then k = 2.5/5 = 0.5 or 5/10 = 0.5
  • If (-3, 6) → (9, -18), then k = 9/(-3) = -3 or (-18)/6 = -3
Quick Tips:
  • Use either x or y coordinates to find the scale factor
  • Always verify with the other coordinate to ensure consistency
  • Scale factor can be positive, negative, or fractional
Frequently Asked Questions:

Q: What if the scale factor is negative?
A: A negative scale factor results in a dilation that is reflected across the center of dilation.

Q: Can I use the distance formula to find scale factor?
A: Yes, k = distance to image / distance to pre-image, but using coordinates is simpler for origin-centered dilations.

Solutions: Exercises 4 to 5
4 Similar figures
Exercise 4
Rectangle PQRS has sides of length 4 and 6. Rectangle WXYZ has sides of length 10 and 15. Are these rectangles similar? If so, what is the scale factor?
Definition:

Similar figures: Two figures are similar if their corresponding angles are equal and their corresponding sides are proportional. For rectangles, this means the ratios of corresponding sides are equal.

Note: For rectangles to be similar, the ratio of length to width must be the same for both rectangles.

Step-by-step similarity determination method:
  1. Identify corresponding sides of both figures
  2. Calculate the ratio of corresponding sides
  3. Check if all ratios are equal
  4. If ratios are equal, the figures are similar and the common ratio is the scale factor
Rectangle PQRS
Sides: 4, 6
Rectangle WXYZ
Sides: 10, 15
Result
Similar, k = 2.5
Step 1: Identify corresponding sides

Assuming corresponding sides are 4 and 10, and 6 and 15

Step 2: Calculate the ratio of corresponding sides

Ratio 1: 10/4 = 2.5

Ratio 2: 15/6 = 2.5

Step 3: Compare the ratios

Since both ratios equal 2.5, the sides are proportional

Step 4: Determine similarity

Since corresponding sides are proportional and all angles are 90°, the rectangles are similar

The scale factor is 2.5

Rectangles PQRS and WXYZ are similar with scale factor k = 2.5
Final answer:

Yes, rectangles PQRS and WXYZ are similar. The scale factor is k = 2.5 (the second rectangle is 2.5 times larger than the first).

Applied rules:

• Similarity definition: Corresponding angles equal and sides proportional

• Rectangle similarity: Ratio of length to width must be equal

• Proportionality check: All corresponding sides must have the same ratio

• Practice Tip: For rectangles, check if (length₁/width₁) = (length₂/width₂)

Related Examples:
  • Rectangles 2×3 and 4×6: (2/4) = (3/6) = 0.5 → Similar
  • Rectangles 3×4 and 6×9: (3/6) = 0.5, (4/9) ≈ 0.44 → Not similar
  • Squares are always similar to each other (equal ratios)
Quick Tips:
  • For rectangles, compare ratios of corresponding dimensions
  • Scale factor can be found by dividing any corresponding lengths
  • Similar figures have the same shape but different sizes
Frequently Asked Questions:

Q: Can rectangles with different orientations be similar?
A: Yes, orientation doesn't affect similarity. A 3×4 rectangle is similar to a 6×8 rectangle regardless of orientation.

Q: Are all rectangles similar to each other?
A: No, only rectangles with the same aspect ratio (length/width) are similar.

5 Composite transformations
Exercise 5
Find the coordinates of the image of point R(4, 2) after a dilation centered at the origin with scale factor 0.5, followed by a translation of 3 units right and 1 unit up.
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: When combining similarity and congruence transformations, the final figure is similar to the original but may be in a different position.

Step-by-step composition method:
  1. Apply the first transformation (dilation) to the original point
  2. Use the result as input for the second transformation (translation)
  3. Continue for all transformations in sequence
  4. Record the final coordinates
Starting Point
R(4, 2)
After Dilation
R'(2, 1)
Final Image
R''(5, 2)
Step 1: Apply first transformation (dilation)

Original point: R(4, 2)

Scale factor: k = 0.5

Dilation rule: (x, y) → (0.5x, 0.5y)

R(4, 2) → R'(0.5×4, 0.5×2) = R'(2, 1)

Step 2: Apply second transformation (translation)

Input: R'(2, 1)

Translation: 3 units right, 1 unit up

Translation rule: (x, y) → (x+3, y+1)

R'(2, 1) → R''(2+3, 1+1) = R''(5, 2)

Step 3: Verify the result

Starting at (4, 2), we ended at (5, 2)

Step 4: Write the final coordinates

The final coordinates after both transformations are R''(5, 2)

After dilation (k=0.5) then translation (3,1), R(4,2) → R''(5,2)
Final answer:

The coordinates of the image of point R(4, 2) after a dilation centered at the origin with scale factor 0.5, followed by a translation of 3 units right and 1 unit up, are R''(5, 2).

Applied rules:

• Dilation rule: (x, y) → (kx, ky)

• Translation rule: (x, y) → (x+h, y+k)

• Composition order: Apply transformations in the specified sequence

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

Related Examples:
  • Point (6, 8) → dilation k=0.5 → (3, 4) → translation (2, -1) → (5, 3)
  • Point (10, 5) → translation (-3, 2) → (7, 7) → dilation k=2 → (14, 14)
  • Order matters: dilating then translating differs from translating then dilating
Quick Tips:
  • Perform transformations in the exact order specified
  • Each transformation's output becomes the next transformation's input
  • Similarity transformations preserve shape but not necessarily position
Frequently Asked Questions:

Q: Does the order of transformations matter when mixing dilations and translations?
A: Yes, the order matters. Dilating then translating usually gives a different result than translating then dilating.

Q: Is the final figure similar to the original when combining transformations?
A: Yes, if at least one dilation is involved, the final figure will be similar to the original.

Solutions: Exercises 6 to 10
6 Proportional reasoning
Exercise 6
Triangle ABC is similar to triangle DEF. If AB = 6 cm, BC = 8 cm, and DE = 9 cm, find the length of EF.
Definition:

Proportional reasoning in similar figures: If two figures are similar, their corresponding sides are proportional. If triangle ABC ~ triangle DEF, then AB/DE = BC/EF = AC/DF.

Note: The ratio of corresponding sides in similar figures is constant and equals the scale factor of the similarity transformation.

Step-by-step proportional reasoning method:
  1. Identify corresponding sides in the similar figures
  2. Set up a proportion using known corresponding sides
  3. Solve for the unknown side length
  4. Verify the solution by checking with other corresponding sides if available
Given Information
AB=6, BC=8, DE=9
Corresponding Sides
AB→DE, BC→EF
Unknown
EF = 12
Step 1: Identify corresponding sides

Since triangle ABC ~ triangle DEF:

AB corresponds to DE

BC corresponds to EF

AC corresponds to DF

Step 2: Set up the proportion

AB/DE = BC/EF

6/9 = 8/EF

Step 3: Solve for EF

Cross multiply: 6 × EF = 9 × 8

6 × EF = 72

EF = 72/6 = 12

Step 4: Verify the solution

Scale factor = AB/DE = 6/9 = 2/3

Check: BC/EF = 8/12 = 2/3 ✓

The length of EF is 12 cm
Final answer:

The length of EF is 12 cm.

Applied rules:

• Similarity proportion: Corresponding sides are proportional in similar figures

• Cross multiplication: Use to solve proportions

• Scale factor consistency: All corresponding sides have the same ratio

• Practice Tip: Always verify your answer by checking with another pair of corresponding sides

Related Examples:
  • Triangles with sides 3,4,5 and 6,8,10: scale factor = 2
  • Similar rectangles: if one has sides 2,3 and scale factor is 4, the other has sides 8,12
  • Proportional reasoning applies to all similar figures
Quick Tips:
  • Identify corresponding sides carefully before setting up proportions
  • Use cross multiplication to solve proportion equations
  • Verify your answer by checking with another pair of corresponding sides
Frequently Asked Questions:

Q: How do I identify corresponding sides in similar figures?
A: Corresponding sides are in the same relative position in each figure and are across from equal angles.

Q: Can I use the scale factor to find missing sides?
A: Yes, once you find the scale factor, you can multiply any original length by it to get the corresponding length in the similar figure.

7 Real-world application
Exercise 7
A map has a scale of 1 inch representing 5 miles. If two cities are 3.5 inches apart on the map, how far are they actually apart? If the actual distance between two other cities is 20 miles, how far apart are they on the map?
Definition:

Scale in real-world applications: A scale represents the ratio between measurements on a model or map and the actual measurements in reality. Scale factors in maps and blueprints are examples of similarity transformations.

Note: The scale provides a constant ratio between corresponding lengths in the model and reality, allowing for proportional reasoning.

Step-by-step scale application method:
  1. Identify the given scale as a ratio
  2. Set up a proportion relating map distance to actual distance
  3. Solve for the unknown quantity
  4. Apply the same principle to the second part of the question
Given Scale
1 in : 5 mi
Part 1
3.5 in → 17.5 mi
Part 2
20 mi → 4 in
Step 1: Set up the scale ratio

Map distance : Actual distance = 1 inch : 5 miles

Step 2: Find actual distance for 3.5 inches on map

1 inch / 5 miles = 3.5 inches / x miles

1/x = 5/3.5

x = 3.5 × 5 = 17.5 miles

Step 3: Find map distance for 20 miles actual

1 inch / 5 miles = y inches / 20 miles

y = 20/5 = 4 inches

Step 4: Present both answers

Two cities 3.5 inches apart on the map are actually 17.5 miles apart

Two cities 20 miles apart in reality are 4 inches apart on the map

3.5 inches on map = 17.5 miles actual; 20 miles actual = 4 inches on map
Final answer:

Two cities that are 3.5 inches apart on the map are actually 17.5 miles apart. Two cities that are 20 miles apart in reality are 4 inches apart on the map.

Applied rules:

• Scale factor: Maintains proportional relationship between model and reality

• Proportional reasoning: Use cross multiplication to solve scale problems

• Unit conversion: Maintain consistency in units when setting up proportions

• Practice Tip: Scale problems always involve proportional relationships

Related Examples:
  • Blueprint scale 1:50: 2cm on blueprint = 100cm actual
  • Model car scale 1:24: 5-inch model = 120-inch actual
  • Photograph enlargement: 4×6 photo enlarged by factor 3 becomes 12×18
Quick Tips:
  • Always write the scale as a ratio (model:actual or map:real)
  • Set up proportions carefully, matching corresponding units
  • Check that your answer makes sense in the real-world context
Frequently Asked Questions:

Q: What's the difference between scale factor and scale ratio?
A: Scale factor is the multiplier (like 0.5), while scale ratio is expressed as a comparison (like 1:20).

Q: How do I convert between different scales?
A: Convert both scales to the same units, then find the ratio between them.

8 Properties of similarity
Exercise 8
Triangle ABC has sides of length 3, 4, and 5. It is dilated with a scale factor of 2 to form triangle A'B'C'. Find the side lengths, perimeter, and area of triangle A'B'C'. Compare these to the original triangle.
Definition:

Properties of similarity transformations: Under a similarity transformation with scale factor k: side lengths change by factor k, perimeter changes by factor k, and area changes by factor k².

Note: Angles remain unchanged in similarity transformations, preserving the shape while altering the size.

Step-by-step properties calculation method:
  1. Calculate new side lengths by multiplying original lengths by scale factor
  2. Calculate new perimeter by multiplying original perimeter by scale factor
  3. Calculate new area by multiplying original area by scale factor squared
  4. Compare the results to understand how each measurement changes
Original Triangle
Sides: 3,4,5; P=12; A=6
Scale Factor
k = 2
Dilated Triangle
Sides: 6,8,10; P=24; A=24
Step 1: Calculate new side lengths

Original sides: 3, 4, 5

New sides: 3×2, 4×2, 5×2 = 6, 8, 10

Step 2: Calculate new perimeter

Original perimeter: 3 + 4 + 5 = 12

New perimeter: 6 + 8 + 10 = 24

Or: 12 × 2 = 24 (perimeter scales by k)

Step 3: Calculate new area

Original area: (1/2) × 3 × 4 = 6 (since it's a right triangle)

New area: (1/2) × 6 × 8 = 24

Or: 6 × 2² = 6 × 4 = 24 (area scales by k²)

Step 4: Compare measurements

Sides: multiplied by k = 2

Perimeter: multiplied by k = 2

Area: multiplied by k² = 4

A'B'C' has sides 6,8,10; perimeter 24; area 24. All linear measures scaled by 2, area by 4.
Final answer:

Triangle A'B'C' has side lengths of 6, 8, and 10 units. The perimeter is 24 units (2 times the original perimeter). The area is 24 square units (4 times the original area of 6 square units). Linear measurements scale by the scale factor k=2, while area scales by k²=4.

Applied rules:

• Linear scaling: Lengths scale by factor k

• Perimeter scaling: Perimeter scales by factor k

• Area scaling: Area scales by factor k²

• Angle preservation: All angles remain unchanged in similarity transformations

Related Examples:
  • Rectangle 2×3 dilated by k=3: new dimensions 6×9, area changes from 6 to 54 (k²=9 times)
  • Circle radius r dilated by k: new radius kr, area changes from πr² to π(kr)² = k²πr²
  • Volume of 3D figures changes by k³ in similarity transformations
Quick Tips:
  • Linear measurements (sides, perimeter) scale by k
  • Area measurements scale by k²
  • Volume measurements scale by k³
Frequently Asked Questions:

Q: Why does area scale by k² instead of k?
A: Area is a two-dimensional measurement, so when linear dimensions are scaled by k, area becomes k×k = k² times the original.

Q: Do angles change during similarity transformations?
A: No, angles remain unchanged in similarity transformations, which is why the shape is preserved.

9 Advanced problem solving
Exercise 9
A dilation centered at point (2, 3) with scale factor 2 maps point M(x, y) to point M'(2x-2, 2y-3). Find the coordinates of point M if M' is at (6, 7).
Definition:

Dilation with center not at origin: When the center of dilation is at point (h, k), the transformation rule is: (x, y) → (h + k(x-h), k + k(y-k)), which simplifies to (x, y) → ((1-k)h + kx, (1-k)k + ky).

Note: For a dilation with center (h, k) and scale factor s, the formula is: (x, y) → (h + s(x-h), k + s(y-k)).

Step-by-step inverse transformation method:
  1. Use the given transformation rule to set up equations
  2. Substitute the known coordinates of the image point
  3. Solve the system of equations to find the original coordinates
  4. Verify the solution by applying the transformation
Given
Center(2,3), k=2, M'=(6,7)
Transformation Rule
(x,y) → (2x-2, 2y-3)
Original Point
M(4, 5)
Step 1: Set up equations using the transformation rule

Given: (x, y) → (2x-2, 2y-3)

We know M'(6, 7), so:

2x - 2 = 6

2y - 3 = 7

Step 2: Solve for x

2x - 2 = 6

2x = 8

x = 4

Step 3: Solve for y

2y - 3 = 7

2y = 10

y = 5

Step 4: Verify the solution

Apply the transformation to M(4, 5):

(2×4 - 2, 2×5 - 3) = (8 - 2, 10 - 3) = (6, 7) ✓

The coordinates of point M are (4, 5)
Final answer:

The coordinates of point M are (4, 5).

Applied rules:

• General dilation rule: With center (h, k) and scale factor s: (x, y) → (h + s(x-h), k + s(y-k))

• Algebraic manipulation: Solve systems of equations to find original coordinates

• Verification: Apply transformation to verify the solution

• Practice Tip: For non-origin centers, use the general dilation formula

Related Examples:
  • Dilation center (1, 2), k=3: (x, y) → (1+3(x-1), 2+3(y-2)) = (3x-2, 3y-4)
  • If (a, b) → (2a+1, 2b-3), then center is (-1, 3) with k=2
  • Reverse transformations require solving algebraic equations
Quick Tips:
  • For dilation with center (h, k), use: (x, y) → (h + s(x-h), k + s(y-k))
  • To find original point from image, solve the transformation equations backwards
  • Always verify your answer by applying the transformation
Frequently Asked Questions:

Q: How do I find the center and scale factor from a transformation rule?
A: Rewrite the rule in the form (x, y) → (ax + b, cy + d) and use algebra to find center and scale factor.

Q: What if the scale factor is negative?
A: The figure is reflected across the center of dilation and scaled by the absolute value of the scale factor.

10 Complex similarity
Exercise 10
Pentagon ABCDE is similar to pentagon FGHIJ. The perimeter of ABCDE is 30 units, and the area is 50 square units. If the scale factor from ABCDE to FGHIJ is 3, find the perimeter and area of pentagon FGHIJ.
Definition:

Scaling properties for any similar figures: For any similar figures with scale factor k: corresponding linear measures (including perimeter) scale by k, areas scale by k², and volumes scale by k³.

Note: These properties apply to figures of any shape, not just triangles or rectangles. The scale factor determines how all measurements change proportionally.

Step-by-step scaling application method:
  1. Identify the scale factor between the similar figures
  2. Apply the scale factor to linear measurements (perimeter)
  3. Apply the square of the scale factor to area measurements
  4. State the results with proper units
Original Pentagon
Perimeter=30, Area=50
Scale Factor
k = 3
Scaled Pentagon
Perimeter=90, Area=450
Step 1: Identify the scale factor

The scale factor from ABCDE to FGHIJ is k = 3

Step 2: Calculate the new perimeter

Linear measurements scale by k

Perimeter of FGHIJ = Perimeter of ABCDE × k

Perimeter of FGHIJ = 30 × 3 = 90 units

Step 3: Calculate the new area

Area measurements scale by k²

Area of FGHIJ = Area of ABCDE × k²

Area of FGHIJ = 50 × 3² = 50 × 9 = 450 square units

Step 4: Present the results

Pentagon FGHIJ has a perimeter of 90 units and an area of 450 square units

FGHIJ has perimeter 90 units and area 450 square units
Final answer:

The perimeter of pentagon FGHIJ is 90 units (3 times the original perimeter), and the area is 450 square units (9 times the original area).

Applied rules:

• Linear scaling: Perimeter scales by factor k

• Area scaling: Area scales by factor k²

• Universal property: Scaling properties apply to figures of any shape

• Practice Tip: Remember: linear by k, area by k², volume by k³

Related Examples:
  • Circle circumference scales by k, area by k²
  • Triangle perimeter scales by k, area by k²
  • Hexagon perimeter scales by k, area by k²
Quick Tips:
  • Linear measurements always scale by k
  • Area measurements always scale by k²
  • These rules apply to all similar figures regardless of shape
Frequently Asked Questions:

Q: Do these scaling properties work for irregular shapes?
A: Yes, the scaling properties apply to any similar figures, whether regular or irregular.

Q: How do I find the scale factor if I only know area ratios?
A: If area ratio is r, then scale factor k = √r (the square root of the area ratio).

Key Laws, Methods, Rules, and Definitions
\((x, y) \rightarrow (kx, ky)\)
Dilation Rule (Origin Center)
Key definitions:

Similarity Transformation: A transformation that preserves shape but not necessarily size. Includes dilations and combinations with rigid transformations.

Dilation: A similarity transformation that enlarges or reduces a figure by a scale factor from a center point.

Scale Factor: The ratio by which corresponding lengths in similar figures are proportional.

Complete methodology:
  1. Identify the transformation type: Determine if it's a dilation or similarity transformation
  2. Apply the appropriate rule: Use the correct transformation rule for coordinates
  3. Calculate the scale factor: If needed, determine the ratio of corresponding sides
  4. Verify similarity: Check that corresponding angles are equal and sides are proportional
Tip 1: Origin-center dilation: (x, y) → (kx, ky) where k is the scale factor.
Tip 2: Linear measures scale by k, areas by k², volumes by k³.
Tip 3: Similar figures have equal corresponding angles and proportional sides.
Tip 4: Scale factor k > 1 enlarges, 0 < k < 1 reduces, k < 0 reflects and scales.
Common errors: Confusing scale factor effects, forgetting area scaling by k², misidentifying corresponding sides.
Exam preparation: Memorize scaling properties, practice proportional reasoning, understand similarity criteria.
Formulas to memorize:

• Origin-center dilation: \((x, y) \rightarrow (kx, ky)\)

• General dilation: \((x, y) \rightarrow (h + s(x-h), k + s(y-k))\) for center \((h, k)\)

• Linear scaling: New measure = k × original measure

• Area scaling: New area = k² × original area

Rules and Methods for Similarity Transformations
\(\frac{\text{New length}}{\text{Original length}} = k\)
Scale Factor Definition
Dilation
\((x, y) \rightarrow (kx, ky)\)
Scale by factor k from origin
Linear Measures
Scaled by k
Lengths, perimeters scale by k
Areas
Scaled by k²
Areas scale by k squared

Key Takeaways

  • Similarity transformations preserve shape but not size
  • Dilations change size by a scale factor from a center point
  • Corresponding angles remain equal in similar figures
  • Corresponding sides are proportional in similar figures
  • Linear measures scale by k, areas by k², volumes by k³

Questions & Answers

Question: How do I know if two figures are similar, and what's the difference from congruent figures?

Answer: Two figures are similar if:

  • Corresponding angles are equal
  • Corresponding sides are proportional (have the same ratio)

The difference from congruent figures:

  • Congruent: Same shape AND same size (scale factor = 1)
  • Similar: Same shape BUT different size (scale factor ≠ 1)

For triangles specifically, there are shortcuts like AA (Angle-Angle), SSS (Side-Side-Side), or SAS (Side-Angle-Side) similarity criteria.

Question: Why does area scale by k² instead of k when figures are dilated?

Answer: Area is a two-dimensional measurement, so it depends on two linear dimensions.

  • If each linear dimension (length and width) is multiplied by the scale factor k
  • Then area = (k × length) × (k × width) = k² × (length × width)
  • So area is multiplied by k²

For example, if you dilate a 2×3 rectangle (area=6) by scale factor 2, you get a 4×6 rectangle (area=24). Notice that 24 = 6 × 2².

Similarly, volume scales by k³ since it's a three-dimensional measurement.

Question: What happens when I combine a dilation with other transformations?

Answer: When you combine a dilation with other transformations (translations, rotations, reflections), the result is still a similarity transformation.

  • The shape of the figure remains the same (angles preserved)
  • The size changes according to the dilation's scale factor
  • The position/orientation may change due to other transformations
  • The final figure is similar to the original (same shape, proportional size)

For example, if you dilate a triangle by factor 2 and then rotate it, the final triangle will be twice as large as the original but in a different orientation. The triangles will still be similar.

Geometry Glossary

Similarity Transformation
A transformation that changes the size of a figure but preserves its shape. Includes dilations and combinations with rigid transformations.
Dilation
A similarity transformation that enlarges or reduces a figure by a scale factor from a center point. Rule: (x, y) → (kx, ky) for center at origin.
Scale Factor
The ratio of corresponding lengths in similar figures. If k > 1, the figure is enlarged; if 0 < k < 1, the figure is reduced.
Similar Figures
Figures that have the same shape but not necessarily the same size. Corresponding angles are equal and corresponding sides are proportional.
Proportional
Having the same ratio. In similar figures, corresponding sides are proportional.
Center of Dilation
The fixed point from which a dilation is performed. All points move toward or away from this point by the scale factor.
Corresponding Parts
Parts (sides or angles) that are in the same relative position in similar figures.

Similarity 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 similarity. All content is aligned with Common Core State Standards and reviewed by mathematics education experts to ensure accuracy and pedagogical effectiveness.