A dilation from the origin with factor k sends (x, y) to (kx, ky).
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Angles do not change
That is why a dilation gives a similar figure rather than a congruent one.
Area scales by the square
Doubling every length multiplies the area by four, because area uses two dimensions.
Factors below one
A factor between 0 and 1 shrinks the figure. It is still a dilation.
The one transformation that changes size
A dilation with centre O and factor k sends each point P to the point on ray OP at k times the distance. Angles are preserved, lengths are scaled. It is the transformation that produces similarity rather than congruence.
Images are parallel to originals
A dilated segment is parallel to the original, because the direction from the centre never changes. That parallelism is the engine behind the side-splitter theorem and most similar-triangle proofs.
What the factor does
k > 1 enlarges, 0 < k < 1 shrinks, k = 1 does nothing, and a negative k enlarges through the centre, producing a rotated-by-180° image. The negative case is often omitted and is worth knowing exists.
Area scales by k², volume by k³
A dilation with factor 3 multiplies areas by 9 and volumes by 27. This is the same squaring and cubing met in Grade 7 and 8, and it explains why models, animals and structures cannot simply be scaled up.
Step 2: Try It Yourself
Tap and try it out.
Every length is multiplied by 2, but every angle is unchanged. The two figures are the same shape at a different size, which is what similar means.
Step 3: Watch an Example
One step at a time.
Watch Kofi Scale an Area
A rectangle of area 12 is dilated by a factor of 3.
- Step 1
Every length is multiplied by 3.
Step 4: Your Turn
Practice makes it stick.
The Point
Problem 1 of 2
(4, 5) dilated by 3 from the origin. New x?
The Area
Problem 2 of 2
Area 5, dilated by factor 2. New area?
Scale It
1 of 8
(3, 7) dilated by 4. New y?
2 of 8
Area 6, factor 3. New area?
3 of 8
Factor 5. By what does area scale?
4 of 8
Does a dilation change angles? 1 for yes, 0 for no.
5 of 8
(8, 6) dilated by ½. New x?
6 of 8
A factor of 1. Is the figure unchanged? 1 for yes, 0 for no.
7 of 8
Volume scales by which power of k? Give the power.
8 of 8
Sort each quantity by how it scales under a dilation of factor k.
Tap something to move it.
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Step 5: Quick Check
Show what you know.
Question 1 of 1
Area 4, factor 3. New area?
What You Learned
- A dilation multiplies both coordinates by the scale factor.
- Angles are unchanged, so the image is similar rather than congruent.
- Lengths scale by k, areas by k², and volumes by k³.