A dilation centred at 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.
Bigger or smaller
A factor above 1 enlarges. A factor between 0 and 1 shrinks.
What survives
Lengths change, but angles do not. That is why a dilation gives a similar figure rather than a congruent one.
Why the origin matters
The rule (kx, ky) only works for a dilation centred at the origin. A different centre needs a different calculation.
Multiply both coordinates
A dilation centred at the origin with factor k sends (x, y) to (kx, ky). Both coordinates are multiplied by the same number, which is what keeps the direction from the origin unchanged.
When the centre is not the origin
Subtract the centre's coordinates, multiply by k, then add the centre back. You are translating the centre to the origin, dilating, and translating back — three simple moves rather than one complicated rule.
Finding the scale factor from coordinates
Divide a coordinate of the image by the matching coordinate of the original, taking the centre as the origin. If different points give different answers, the transformation was not a dilation.
Check the image looks right
A factor of 2 should give a figure visibly twice as far from the centre in every direction. Plotting a couple of points and eyeballing the result catches an arithmetic slip faster than rechecking the algebra.
Step 2: Try It Yourself
The original stays, drawn dashed, so the change in size is visible.
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 Ana Dilate a Corner
A corner sits at (3, −2) and the figure is dilated by 4 from the origin.
- Step 1
She multiplies the x by the factor: 3 × 4 = 12.
Step 4: Your Turn
Practice makes it stick.
The Enlargement
Problem 1 of 2
(3, 5) dilated by 2 from the origin. What is the new x?
The Reduction
Problem 2 of 2
(8, 4) dilated by ½ from the origin. What is the new y?
Multiply Both
1 of 4
(2, 7) dilated by 3. What is the new y?
2 of 4
(10, 6) dilated by ½. What is the new x?
3 of 4
Does a dilation change the angles? 1 for yes, 0 for no.
4 of 4
A factor of 1. How many times bigger is the figure?
Step 5: Quick Check
Show what you know.
Question 1 of 1
(4, −3) dilated by 5. What is the new y?
What You Learned
- A dilation from the origin multiplies both coordinates by the scale factor.
- Above 1 enlarges, between 0 and 1 shrinks.
- Angles are unchanged, which is why the result is similar rather than congruent.