Solved Exercises on Dilations in Grade 8

Master dilations: scale factor, center of dilation, geometric transformations, and applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Enlargement with Scale Factor 2
Exercise 1
Dilate point A(3, 4) with a scale factor of 2 centered at the origin. What are the coordinates of the image point A'?
Definition:

Dilation: A geometric transformation that changes the size of a figure by a scale factor from a fixed point called the center of dilation

Scale Factor: The ratio of the size of the image to the size of the original figure

Rule for Dilation at Origin: If point (x, y) is dilated by scale factor k, the image is (kx, ky)

Dilation Method:
  1. Identify the original point coordinates (x, y)
  2. Identify the center of dilation and scale factor k
  3. Apply the dilation rule: (x, y) → (kx, ky)
  4. Calculate the new coordinates
  5. Verify the result by checking distance ratios
Original Point
A(3, 4)
Scale Factor
k = 2
Image Point
A'(6, 8)
Step 1: Identify Original Coordinates

Point A has coordinates (3, 4), so x = 3 and y = 4

Step 2: Identify Scale Factor

Scale factor k = 2 (enlargement since k > 1)

Step 3: Apply Dilation Rule

When dilating at origin with scale factor k, (x, y) → (kx, ky)

So (3, 4) → (2×3, 2×4) = (6, 8)

Step 4: Write Image Coordinates

The image point A' has coordinates (6, 8)

Step 5: Verify Result

Distance from origin: A(3,4) = √(3² + 4²) = 5, A'(6,8) = √(6² + 8²) = 10

Ratio: 10/5 = 2 = scale factor ✓

A'(6, 8)
Final answer:

The coordinates of the image point A' are (6, 8)

Applied rules:

• Dilation Rule: (x, y) → (kx, ky) when center is at origin

• Scale Factor: k > 1 means enlargement, 0 < k < 1 means reduction

• Distance Ratio: Distance from center of dilation increases by factor k

2 Reduction with Fractional Scale Factor
Exercise 2
Triangle ABC has vertices A(6, 8), B(4, 2), and C(8, 4). Dilate the triangle with a scale factor of 1/2 centered at the origin. Find the coordinates of A'B'C'.
Definition:

Reduction: A dilation where the scale factor is between 0 and 1, making the figure smaller

Properties Preserved: Shape, angle measures, and orientation remain unchanged

Proportional Scaling: All distances from center of dilation change by the same factor

Original Triangle
A(6,8), B(4,2), C(8,4)
Scale Factor
k = 1/2
Dilated Triangle
A'(3,4), B'(2,1), C'(4,2)
Step 1: Apply Dilation to Vertex A

A(6, 8) → A'(1/2×6, 1/2×8) = A'(3, 4)

Both coordinates multiplied by 1/2

Step 2: Apply Dilation to Vertex B

B(4, 2) → B'(1/2×4, 1/2×2) = B'(2, 1)

Both coordinates multiplied by 1/2

Step 3: Apply Dilation to Vertex C

C(8, 4) → C'(1/2×8, 1/2×4) = C'(4, 2)

Both coordinates multiplied by 1/2

Step 4: Verify Triangle Properties

The dilated triangle A'B'C' has the same shape as the original triangle ABC

All distances from origin are reduced by factor of 1/2

A'(3, 4), B'(2, 1), C'(4, 2)
Final answer:

The coordinates of the dilated triangle A'B'C' are A'(3, 4), B'(2, 1), and C'(4, 2)

Applied rules:

• Dilation Rule: (x, y) → (kx, ky) when center is at origin

• Apply to Each Vertex: Rule applied individually to each vertex

• Preservation: Shape, angles, and orientation preserved

3 Dilation with Different Center
Exercise 3
Point P(5, 3) is dilated with a scale factor of 3 centered at point C(2, 1). Find the coordinates of the image point P'.
Definition:

Dilation Around Arbitrary Point: To dilate around a center that is not the origin, first translate the center to the origin, perform the dilation, then translate back

Formula: P' = C + k(P - C) where C is center, P is original point, k is scale factor

Original Point
P(5, 3)
Center of Dilation
C(2, 1)
Scale Factor
k = 3
Step 1: Translate Center to Origin

Translate by (-2, -1) to move C(2, 1) to (0, 0)

P(5, 3) → P'(5-2, 3-1) = P'(3, 2)

Step 2: Apply Dilation with Scale Factor 3

P'(3, 2) → P''(3×3, 3×2) = P''(9, 6)

Step 3: Translate Back

Translate by (2, 1) to move center back to C(2, 1)

P''(9, 6) → P'(9+2, 6+1) = P'(11, 7)

Step 4: Calculate Using Formula

Using P' = C + k(P - C)

P' = (2, 1) + 3[(5, 3) - (2, 1)]

P' = (2, 1) + 3(3, 2) = (2, 1) + (9, 6) = (11, 7)

Step 5: Verify Result

Distance from C to P: √[(5-2)² + (3-1)²] = √[9 + 4] = √13

Distance from C to P': √[(11-2)² + (7-1)²] = √[81 + 36] = √117 = 3√13

Ratio: (3√13)/√13 = 3 ✓

P'(11, 7)
Final answer:

The coordinates of the image point P' are (11, 7)

Applied rules:

• Translation Method: Move center to origin, dilate, then move back

• Formula: P' = C + k(P - C)

• Distance Preservation: Distance ratios remain constant

Rules and methods, laws,...
\((x, y) \rightarrow (kx, ky)\)
Dilation at Origin
Dilation at Origin
\((x, y) \rightarrow (kx, ky)\)
Scale factor k
Dilation at Point C
\(P' = C + k(P - C)\)
Arbitrary center
Scale Factor Effect
k > 1: Enlargement, 0 < k < 1: Reduction
Size change
Key definitions:

Preimage: The original figure before transformation

Image: The figure after transformation

Similar Figures: Figures with the same shape but different sizes (dilations produce similar figures)

Center of Dilation: The fixed point from which the dilation occurs

Complete methodology:
  1. Identify Preimage: Locate original points, lines, or shapes
  2. Determine Center and Scale Factor: Find the center of dilation and the scale factor k
  3. Apply Dilation Rule: Use the appropriate rule for the center and scale factor
  4. Calculate New Coordinates: Perform the transformation for each point
  5. Draw Image: Plot the dilated figure
  6. Verify Properties: Check that shape is preserved and size changes by scale factor
Tip 1: Scale factor > 1 means enlargement, 0 < scale factor < 1 means reduction.
Tip 2: For dilations at origin, multiply each coordinate by the scale factor.
Tip 3: For dilations at other centers, use the translate-dilate-translate-back method.
Tip 4: The center of dilation remains fixed (does not move during dilation).
Common errors: Forgetting to apply scale factor to both coordinates, applying dilation to the wrong center, confusing enlargement with reduction.
Exam preparation: Practice with different scale factors (including fractions), understand that angles are preserved but lengths change by scale factor, and verify that figures remain similar after dilation.
Formulas to know by heart:

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

• Dilation at center C: P' = C + k(P - C)

• Enlargement: k > 1

• Reduction: 0 < k < 1

• Distance ratio: Distance from center of dilation changes by factor k

Exercise with Visualization: Dilation Properties
Exercise 6: Dilation Effects on Shapes
Examine how dilations affect distances and shapes:
Original triangle: A(0,0), B(3,0), C(0,4)
Dilated triangle (scale factor 2): A'(0,0), B'(6,0), C'(0,8)
Verify that shapes are similar and distances change by scale factor.

Analysis: The chart shows how dilations change distances while preserving shape.

  • Original triangle: AB = 3, AC = 4, BC = 5 (3-4-5 right triangle)
  • Dilated triangle: A'B' = 6, A'C' = 8, B'C' = 10 (same 3-4-5 ratio)
  • All sides doubled (scale factor 2), confirming similarity!

Questions & Answers

Question: What's the difference between a dilation and other transformations like translation, rotation, or reflection?

Answer: The key differences are:

  • Dilation: Changes the size of the figure while keeping the same shape. The distance from the center of dilation changes by the scale factor.
  • Translation: Moves the figure without changing size or orientation.
  • Rotation: Turns the figure around a point without changing size.
  • Reflection: Flips the figure over a line without changing size.

Translations, rotations, and reflections are rigid transformations (isometries) that preserve distance and angle measures. Dilations are not rigid transformations - they preserve angles but change distances by the scale factor.

Think of dilation like zooming in/out on a photo (size changes, shape preserved), while the others are like moving, turning, or flipping without resizing.

Question: How do I determine if a dilation is an enlargement or a reduction?

Answer: The scale factor determines the type of dilation:

  • Enlargement: When the scale factor k > 1 (figure gets bigger)
  • Reduction: When 0 < k < 1 (figure gets smaller)
  • No Change: When k = 1 (figure stays the same size)

For example:

  • k = 2, 3, 1.5 → Enlargements
  • k = 0.5, 0.25, 0.75 → Reductions
  • k = -2 → Enlargement with reflection (negative scale factors create mirror images)

The absolute value of the scale factor determines the size change, while the sign determines if there's also a reflection.

If |k| > 1, it's an enlargement; if 0 < |k| < 1, it's a reduction.

Question: What properties of a figure stay the same after dilation? What changes?

Answer: In a dilation, the following properties STAY THE SAME:

  • Shape of the figure
  • All angle measures
  • Parallelism (parallel lines remain parallel)
  • Perpendicularity (perpendicular lines remain perpendicular)
  • Orientation of the figure

What CHANGES:

  • Size of the figure (all lengths change by the scale factor)
  • Distance from any point to the center of dilation (changes by scale factor)
  • Area of the figure (changes by scale factor squared)

This is why dilations create similar figures - they have the same shape but different sizes. The figures are proportional to each other.

For example, if you dilate a square with side length 2 by scale factor 3, you get a square with side length 6. Both are squares (same shape), but the second is 3 times larger in each dimension.

Question: Where are dilations used in real life? It seems like an abstract concept.

Answer: Dilations have many practical applications:

  1. Maps and Blueprints: Creating scaled versions of real-world objects
  2. Photography: Zooming in and out, resizing images
  3. Architecture: Building models and creating construction plans
  4. Engineering: Designing parts that need to be scaled up or down
  5. Medicine: Enlarging medical images for diagnosis
  6. Cartography: Creating maps at different scales
  7. Computer Graphics: Resizing and scaling objects in design software

When you see a map with a scale like "1 inch = 10 miles," that's a dilation! The mathematical concept models this common real-world scaling process.

In manufacturing, parts are often designed at one scale and then produced at a different scale using dilation principles.

Question: How can I verify that a dilation produced a similar figure? What should I check?

Answer: To verify similarity after dilation, check these properties:

  1. Angle Measures: Corresponding angles should be equal
  2. Proportional Sides: Corresponding sides should be in the same ratio (scale factor)
  3. Distance Ratios: Distances from center of dilation should be in the same ratio
  4. Shape: The overall shape should be preserved

For example, if you dilated triangle ABC to A'B'C' with scale factor 2:

  • Verify: ∠A = ∠A', ∠B = ∠B', ∠C = ∠C'
  • Verify: A'B'/AB = B'C'/BC = A'C'/AC = 2 (scale factor)
  • Verify: Distance from center to A'/Distance from center to A = 2

You can also verify by checking that the ratio of any corresponding linear measurement equals the scale factor.

Remember: Dilations always produce similar figures by definition, so if all corresponding sides are proportional and angles are equal, the figures are similar!