A dilation multiplies every distance from a centre by the same scale factor.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Lengths change
A factor of 3 makes every side three times longer. This is not a rigid motion.
Angles do not
Every angle survives a dilation untouched, which is what keeps the shape recognisable.
A factor under 1 shrinks
A factor of 1/2 halves every length. It is still a dilation.
The move that changes size
A dilation scales a figure from a fixed point called the centre. Every point moves along the ray from the centre, ending up a scale factor times as far away. Lengths change; angles do not.
What the scale factor does
A factor greater than 1 enlarges, a factor between 0 and 1 shrinks, and a factor of exactly 1 leaves the figure alone. The centre itself never moves, whatever the factor — it is the one fixed point of the transformation.
Dilations are not rigid motions
Because lengths change, a dilation is not a rigid motion and does not produce a congruent figure. It produces a similar one. This is the distinction the whole chapter rests on.
Sides stay parallel
A dilated segment is parallel to the original — the direction never changes, only the distance from the centre. That parallelism is what makes dilations useful in proofs about similar triangles.
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 Ana Dilate a Triangle
A triangle with sides 3, 4, and 5 is dilated by 2.
- Step 1
She multiplies every side by 2, giving 6, 8, and 10.
Step 4: Your Turn
Practice makes it stick.
The Enlargement
Problem 1 of 2
A side of 7 is dilated by 3. How long is it now?
The Angle
Problem 2 of 2
An angle of 40 degrees is dilated by 5. What is it now?
degrees
Scale It
1 of 4
A side of 5 dilated by 4. New length?
2 of 4
A side of 12 dilated by 1/2. New length?
3 of 4
An angle of 75 dilated by 2. What is it now?
4 of 4
Is a dilation a rigid motion?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A side of 9 dilated by 3. New length?
Question 2 of 2
What does a dilation leave unchanged?
What You Learned
- A dilation multiplies every length by a scale factor.
- Angles are unchanged.
- It is not a rigid motion, because lengths change.