Solved Exercises on Dilations in High School Geometry

Master dilations: scale factors, coordinate transformations, similarity properties, and geometric transformations through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic dilation with origin center
Exercise 1
Triangle ABC has vertices A(2, 3), B(4, 1), C(1, -2). Find the coordinates of the image triangle A'B'C' after a dilation centered at the origin with scale factor k = 3.
Definition:

Dilation: A transformation that changes the size of a figure while preserving its shape. The rule for dilation centered at origin with scale factor k is: (x, y) → (kx, ky)

Dilation method:
  1. Identify the center of dilation (origin in this case)
  2. Apply the dilation rule (x, y) → (kx, ky) to each vertex
  3. Multiply each coordinate by the scale factor
  4. Plot the new points and connect them
Original Points
A(2,3), B(4,1), C(1,-2)
Scale Factor
k = 3
Transformed Points
A'(6,9), B'(12,3), C'(3,-6)
Step 1: Apply dilation rule to point A

A(2,3) → A'(3×2, 3×3) = A'(6,9)

Step 2: Apply dilation rule to point B

B(4,1) → B'(3×4, 3×1) = B'(12,3)

Step 3: Apply dilation rule to point C

C(1,-2) → C'(3×1, 3×(-2)) = C'(3,-6)

Step 4: Verify the transformation

Since k > 1, the figure is enlarged. Distances from origin increase by factor of 3.

A'(6,9), B'(12,3), C'(3,-6)
Final answer:

The image triangle A'B'C' has vertices A'(6,9), B'(12,3), C'(3,-6)

Applied rules:

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

• Scale Factor: k > 1 enlarges, k < 1 reduces, k < 0 reflects

• Similarity: Original and image are similar triangles

2 Dilation with non-origin center
Exercise 2
Point P(5, 7) is dilated with center C(2, 3) and scale factor k = 0.5. Find the coordinates of the image point P'.
Definition:

Dilation with center C(h,k): The rule is: (x, y) → (h + k(x-h), k + k(y-k)), where (h,k) is the center and k is the scale factor

Original Point
P(5,7)
Center
C(2,3)
Scale Factor
k = 0.5
Image Point
P'(3.5, 5)
Step 1: Translate to move center to origin

Translate by (-2, -3): P(5,7) → P'(3,4)

Step 2: Apply dilation at origin

P'(3,4) → P''(0.5×3, 0.5×4) = P''(1.5, 2)

Step 3: Translate back

P''(1.5, 2) → P'''(1.5+2, 2+3) = P'(3.5, 5)

Step 4: Verify using direct formula

x' = 2 + 0.5(5-2) = 2 + 1.5 = 3.5

y' = 3 + 0.5(7-3) = 3 + 2 = 5

P'(3.5, 5)
Final answer:

The image point P' has coordinates (3.5, 5)

Applied rules:

• Non-Origin Center: (x, y) → (h + k(x-h), k + k(y-k))

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

• Scale Factor: k < 1 reduces the distance from center

3 Finding scale factor from preimage and image
Exercise 3
Rectangle ABCD with vertices A(1, 2), B(4, 2), C(4, 5), D(1, 5) is dilated to rectangle A'B'C'D' with vertices A'(3, 6), B'(12, 6), C'(12, 15), D'(3, 15). Find the center of dilation and scale factor.
Definition:

Scale Factor: The ratio of distances from center to image points over distances from center to preimage points. k = distance(image)/distance(preimage)

Preimage
ABCD
Image
A'B'C'D'
Result
Center: (0,0), k=3
Step 1: Set up equations using dilation formula

For point A: A'(3,6) = (h + k(1-h), k + k(2-k))

For point B: B'(12,6) = (h + k(4-h), k + k(2-k))

Step 2: Solve for center (h,k)

From A: 3 = h + k(1-h) and 6 = k + k(2-k)

From B: 12 = h + k(4-h) and 6 = k + k(2-k)

Solving gives h = 0, k = 0 (center is origin)

Step 3: Calculate scale factor

Since center is origin: A(1,2) → A'(3,6)

k = 3/1 = 3 (x-coordinate ratio) and k = 6/2 = 3 (y-coordinate ratio)

Step 4: Verify with other points

B(4,2) → B'(12,6): 12/4 = 3 and 6/2 = 3 ✓

C(4,5) → C'(12,15): 12/4 = 3 and 15/5 = 3 ✓

Center: (0,0), Scale Factor: k = 3
Final answer:

The dilation has center (0,0) and scale factor k = 3

Applied rules:

• Scale Factor Calculation: Ratio of corresponding distances

• Center Identification: Point that maps to itself

• Consistency Check: Scale factor should be same for all points

Dilation Rules, Laws and Methods
\((x, y) \rightarrow (kx, ky)\)
Dilation at Origin
Rule 1
\((x, y) \rightarrow (kx, ky)\)
Dilation at origin with scale factor k
Rule 2
\((x, y) \rightarrow (h + k(x-h), k + k(y-k))\)
Dilation with center (h,k) and scale factor k
Rule 3
k > 1: enlargement, k < 1: reduction
Effect of scale factor on size
Key definitions:

Dilation: A transformation that changes the size of a figure while preserving its shape

Scale Factor: The ratio of distances from center to image points over distances from center to preimage points

Center of Dilation: The fixed point that remains unchanged under the transformation

Complete methodology:
  1. Identify the center: Determine if center is origin or another point
  2. Identify the scale factor: Determine the value of k
  3. Apply the transformation: Use the appropriate dilation rule
  4. Verify the result: Check that distances scale by the factor k
Tip 1: When the center is not the origin, translate to origin, dilate, then translate back.
Tip 2: Similar figures have equal angles and proportional sides.
Tip 3: Negative scale factors create reflections across the center.
Tip 4: Always verify your answer by checking the scale factor with multiple points.
Properties of Dilations: Preserves angle measures, parallelism, and orientation (if k > 0).
Common Errors: Forgetting to adjust for non-origin centers, miscalculating scale factors.
Formulas to know by heart:

• Dilation at origin: (x,y) → (kx,ky)

• Dilation at center (h,k): (x,y) → (h+k(x-h), k+k(y-k))

• Scale factor: k = distance(image)/distance(original)

• Area scaling: Area(image) = k² × Area(original)

• Perimeter scaling: Perimeter(image) = |k| × Perimeter(original)

Solution: Exercises 4 to 5
4 Dilation of a circle
Exercise 4
Circle with center (2, 4) and radius 3 is dilated with center of dilation at (2, 4) and scale factor k = 2. Find the equation of the image circle.
Definition:

Dilation of a Circle: When a circle is dilated about its center, only the radius changes by the scale factor.

Original Circle
(x-2)² + (y-4)² = 9
Scale Factor
k = 2
Image Circle
(x-2)² + (y-4)² = 36
Step 1: Identify original parameters

Center: (2, 4), Radius: r = 3

Step 2: Apply dilation to radius

New radius = k × original radius = 2 × 3 = 6

Step 3: Write equation of image circle

Center remains (2, 4), radius is now 6

Equation: (x-2)² + (y-4)² = 6² = 36

Step 4: Verify the result

Area of original: π × 3² = 9π

Area of image: π × 6² = 36π

Ratio: 36π/9π = 4 = k² ✓

(x-2)² + (y-4)² = 36
Final answer:

The equation of the image circle is (x-2)² + (y-4)² = 36

Applied rules:

• Circle Dilation: Only radius changes if center is dilation center

• Area Scaling: Area scales by k²

• Radius Scaling: Radius scales by |k|

5 Real-world application
Exercise 5
A rectangular garden measuring 10 ft by 15 ft is being redesigned. The new garden will be a dilation of the original with scale factor k = 1.5. Calculate the area of the new garden and determine how much more fencing is needed.
Definition:

Real-World Applications: Dilations model scaling in architecture, engineering, maps, and blueprints.

Original Garden
10ft × 15ft
Scale Factor
k = 1.5
New Dimensions
15ft × 22.5ft
Step 1: Calculate original dimensions and area

Length = 15 ft, Width = 10 ft

Area = 15 × 10 = 150 sq ft

Perimeter = 2(15 + 10) = 50 ft

Step 2: Apply scale factor to dimensions

New length = 1.5 × 15 = 22.5 ft

New width = 1.5 × 10 = 15 ft

Step 3: Calculate new area and perimeter

New area = 22.5 × 15 = 337.5 sq ft

New perimeter = 2(22.5 + 15) = 75 ft

Step 4: Calculate additional fencing needed

Additional fencing = 75 - 50 = 25 ft

New area: 337.5 sq ft, Additional fencing: 25 ft
Final answer:

The new garden has area 337.5 sq ft and requires 25 ft more fencing than the original.

Applied rules:

• Linear Scaling: Lengths scale by factor k

• Area Scaling: Areas scale by factor k²

• Perimeter Scaling: Perimeters scale by factor k

Key Concepts: Properties and Relationships
\(\text{Area}_{new} = k^2 \times \text{Area}_{original}\)
Area Scaling Property
Key definitions:

Similarity: Two figures are similar if one is a dilation of the other

Proportional Sides: In similar figures, corresponding sides are proportional

Equal Angles: In similar figures, corresponding angles are equal

Complete methodology:
  1. Analyze the figure: Identify the center of dilation and scale factor
  2. Choose the method: Apply appropriate dilation rule based on center location
  3. Transform each point: Apply the dilation rule to each vertex or significant point
  4. Verify the result: Check that distances scale by the factor k
Tip 1: For dilations with center not at origin, remember the translation steps.
Tip 2: When scale factor is between 0 and 1, the figure shrinks.
Tip 3: Negative scale factors create reflections across the center.
Tip 4: Always check that the center remains fixed in the transformation.
Properties Preserved: Angle measures, parallelism, collinearity.
Properties Changed: Side lengths, perimeter, area, distance from center.
Formulas to know by heart:

• Dilation at origin: (x,y) → (kx,ky)

• Dilation at center (h,k): (x,y) → (h+k(x-h), k+k(y-k))

• Distance scaling: d' = |k| × d

• Area scaling: A' = k² × A

• Perimeter scaling: P' = |k| × P

Exercise with Visualization: Dilation Properties
Exercise 6: Dilation Analysis
Consider the effect of different scale factors on a triangle:
Triangle with vertices A(1,1), B(3,1), C(2,3)
Compare dilations with scale factors k = 0.5, k = 2, k = -1

Analysis: The chart shows how different scale factors affect the size and position of the triangle.

  • k = 0.5: Reduces the triangle by half (shrink)
  • k = 2: Doubles the size of the triangle (enlargement)
  • k = -1: Reflects the triangle across the center (same size, opposite direction)

Questions & Answers

Question: I'm confused about when the scale factor is negative. What does it mean to have a negative scale factor in dilations?

Answer: Great question! A negative scale factor creates what's called a "negative dilation" which combines scaling with a reflection:

  • The absolute value of the scale factor determines the size change (just like positive scale factors)
  • The negative sign means the image appears on the opposite side of the center of dilation
  • Essentially, a negative dilation is equivalent to a positive dilation followed by a reflection through the center

For example, if you dilate point A(2,3) with center (0,0) and scale factor k = -2:

  • First apply the scale: (2×2, 2×3) = (4, 6)
  • Then reflect through origin: (-4, -6)

So A(2,3) → A'(-4, -6) under dilation with center (0,0) and k = -2.

Question: How do I find the center of dilation if I'm only given a preimage and image? It seems impossible without knowing it beforehand.

Answer: Actually, finding the center of dilation from preimage and image is quite systematic! Here's how:

  • Draw lines connecting corresponding points (like A to A')
  • All these connecting lines will intersect at the center of dilation
  • Alternatively, use the dilation formula: if (x,y) → (x',y'), then x' = h + k(x-h) and y' = k + k(y-k)
  • You can solve these equations using coordinates of two corresponding points

For example, if A(1,2) → A'(3,6) and B(4,2) → B'(12,6), set up equations:

3 = h + k(1-h) and 6 = k + k(2-k)

12 = h + k(4-h) and 6 = k + k(2-k)

Solving gives h = 0, k = 0, so center is (0,0).

Question: Why do areas scale by k² instead of just k like the lengths? This doesn't make sense to me.

Answer: This is a common source of confusion! Think of it this way:

  • Area is measured in square units (length × width)
  • If length scales by factor k, then both dimensions scale by k
  • So area scales by k × k = k²

For example, if you have a rectangle 2×3 = 6 sq units, and you dilate with k=2:

  • New dimensions: 4×6 = 24 sq units
  • Original area was 6, new area is 24
  • 24/6 = 4 = 2² = k²

This works for any shape because area involves multiplying two linear dimensions, each scaled by k.

Similarly, volume would scale by k³ since it involves three dimensions.

Question: How can I quickly identify if two figures are related by a dilation? Are there specific visual cues?

Answer: Yes, there are several visual cues to identify dilations:

  • Same shape, different size: The figures are similar (same angles, proportional sides)
  • Parallel corresponding sides: If one is a dilation of the other, corresponding sides are parallel
  • Lines through corresponding vertices meet at one point: Draw lines from corresponding vertices - they should all pass through the center of dilation
  • Constant ratio: Measure distances from center to corresponding points - the ratio should be constant

For example, if you have two triangles, measure corresponding sides. If their ratios are all equal (say 2:1), then one is a dilation of the other with scale factor 2.

Remember that rotations and translations don't change size, so if sizes are different, it's likely a dilation.

Question: What happens to circles, lines, and other geometric figures under dilation? Do they behave differently than polygons?

Answer: Different geometric figures do have distinct behaviors under dilation:

  • Circles: Transform to circles with radius multiplied by |k|. If center of dilation is the center of the circle, only the radius changes
  • Lines: Transform to parallel lines (unless the line passes through the center of dilation, in which case it maps to itself)
  • Triangles/Polygons: Transform to similar figures with proportional sides and equal angles
  • Angles: Always preserved regardless of the figure

For example:

  • A circle with center (h,k) and radius r becomes a circle with center (h,k) and radius |k|×r when dilated from its own center
  • A line passing through the center of dilation remains unchanged
  • A line not passing through the center maps to a parallel line

The fundamental principle is that dilations preserve shape while changing size, maintaining the same geometric relationships.