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.
Know these first
This unlocks
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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