Dilations on the Coordinate Plane
A dilation makes a figure bigger or smaller without changing its shape. Every length is multiplied by the same scale factor, and every angle stays the same.
What you'll be able to do
- Dilate points and figures about the origin with (x, y) → (kx, ky).
- Find the scale factor from a figure and its image.
- Use the scale factor to find a missing length.
- Explain why area changes by k² when lengths change by k.
The big idea
A dilation about the origin multiplies both coordinates by the scale factor k: (x, y) → (kx, ky).
If k is greater than 1, the figure grows (an enlargement). If k is between 0 and 1, it shrinks (a reduction). Angles never change, so the image has the same shape - it's similar, not congruent.
Lengths get multiplied by k, but areas get multiplied by k², because area uses two lengths. Triple every side and the area becomes 9 times as big.
Words you need
- Scale factor (k)
- The number every length is multiplied by.
- Enlargement
- A dilation with k > 1.
- Reduction
- A dilation with 0 < k < 1.
- Center of dilation
- The fixed point the figure grows from or shrinks toward - here, usually the origin.
Worked examples
Example 1
Dilate the point (4, 6) about the origin by a scale factor of 1/2.
- Multiply each coordinate by 1/2.
- (4 × 1/2, 6 × 1/2) = (2, 3).
Answer: (2, 3)
Example 2
Dilate triangle ABC with A(1, 2), B(3, 2) and C(1, 4) about the origin by a scale factor of 3.
- Multiply every coordinate by 3.
- A′(3, 6), B′(9, 6), C′(3, 12).
Answer: (3, 6), (9, 6), (3, 12)
Example 3
A shape with area 6 is dilated by a scale factor of 2. What is the area of the image?
- Lengths double, so area grows by 2² = 4.
- 6 × 4 = 24.
Answer: 24
Common mistakes
Multiplying area by k. Area uses two lengths, so it changes by k × k = k².
Adding the scale factor instead of multiplying. A dilation multiplies: (x, y) → (kx, ky). Adding is a translation.
Calling a dilated figure congruent. A dilation changes size, so the image is similar, not congruent (unless k = 1).
Practice: 20 questions
13 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Dilate the point (3, 5) about the origin by a scale factor of 2.
- Dilate the point (8, −4) about the origin by a scale factor of 1/2.
- A photo's corner at (4, 3) is enlarged about the origin by a scale factor of 3. Where is the corner now? Word problem
- A logo's point at (10, 6) is shrunk about the origin by a scale factor of 1/2. Where is the point now? Word problem
- A map point at (2, −1) is scaled up about the origin by a factor of 4. Where is it now? Word problem
Tier 2 Intermediate One more step to think through
- Point (2, 3) maps to (6, 9), and point (1, 0) maps to (3, 0). What dilation about the origin is this?
- Dilate triangle ABC with A(2, 2), B(4, 2) and C(2, 6) about the origin by a scale factor of 1/2.
- A triangle with sides 3, 4 and 5 is dilated to a triangle with sides 9, 12 and 15. What is the scale factor?
- A poster is 24 inches by 18 inches. A drawing of it is 6 inches by 4.5 inches. What scale factor turns the poster into the drawing? Word problem
- A point at (8, −4) on a blueprint lands at (2, −1) after a dilation about the origin. What is the scale factor? Word problem
- A photo with an area of 24 square inches is enlarged by a scale factor of 3. What is the area of the enlargement? Word problem
- A triangle whose longest side is 5 cm is dilated by a scale factor of 2. How long is the image's longest side? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- Point (2, 4) maps to (3, 6), and point (4, 0) maps to (6, 0). What dilation about the origin is this?
- A garden plan has an area of 80 square feet. The plan is redrawn with a scale factor of 1/2. What is the new area? Word problem
- A photo is 5 inches by 7 inches. It is enlarged so the short side is 15 inches. How long is the long side? Word problem
- A point at (2, 1) is dilated about the origin by a scale factor of 3 and then slid 2 units left. Where does it end up? Word problem
- After a dilation with a scale factor of 3, which stay the same: the angles, the side lengths, or both? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- Dilate the point (3, 2) by a scale factor of 2 about the point (1, 1). Above grade level
- A map is enlarged so its area becomes 9 times as big. What scale factor was used? Word problem
- A projector sends the point (−2, 4) to (−5, 10) and the point (2, 0) to (5, 0). What is the scale factor of this dilation about the origin? Word problem
Answers
Tier 1 Foundations
- (6, 10). Apply (x, y) → (2x, 2y) to each point.
- (4, −2). Apply (x, y) → (0.5x, 0.5y) to each point.
- (12, 9). Apply (x, y) → (3x, 3y) to each point.
- (5, 3). Apply (x, y) → (0.5x, 0.5y) to each point.
- (8, −4). Apply (x, y) → (4x, 4y) to each point.
Tier 2 Intermediate
- (x, y) → (3x, 3y). Apply the rule to every original point; each one lands on its image point.
- (1, 1), (2, 1), (1, 3). Apply (x, y) → (0.5x, 0.5y) to each point.
- Scale factor 3. Each side of the image is 3 times its partner: 9 ÷ 3 = 3.
- Scale factor 0.25. Each side of the image is 0.25 times its partner: 6 ÷ 24 = 0.25.
- (x, y) → (0.25x, 0.25y). Apply the rule to every original point; each one lands on its image point.
- 216. Area scales by k² = 3² = 9, so the new area is 24 × 9 = 216.
- 10 cm. Every length is multiplied by the scale factor: 5 × 2 = 10 cm.
Tier 3 Advanced
- (x, y) → (1.5x, 1.5y). Apply the rule to every original point; each one lands on its image point.
- 20. Area scales by k² = 0.5² = 0.25, so the new area is 80 × 0.25 = 20.
- 21. Scale factor = 15 ÷ 5 = 3, so the missing side is 7 × 3 = 21.
- (4, 3). Apply (x, y) → (3x, 3y); then (x, y) → (x − 2, y) to each point.
- Only the angles. A dilation keeps the shape, so every angle stays the same, but every side length is multiplied by 3.
Tier 4 Challenge
- (5, 3). Apply dilate by 2 about (1, 1) to each point.
- 3. Area grows by k², so k² = 9 and k = 3.
- (x, y) → (2.5x, 2.5y). Apply the rule to every original point; each one lands on its image point.
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