Dilations & Similarity

A dilation multiplies every distance from one fixed point by the same number. Angles survive; lengths do not.

A projector throws a small slide onto a big wall. The picture on the wall is larger. Is it a different picture?

The hill is still a hill. The sun is still round. Nothing got bent.

a projector makes a bigger copy, never a different shape lamp the slide the picture on the wall twice as far from the lamp means twice as wide and twice as tall

Move the wall twice as far from the lamp and the picture becomes twice as wide and twice as tall.

Every length doubled. But the corner where the roof meets the wall is still the same corner — angles did not change at all.

That is the difference between a dilation and every move in the last unit: distances change, angles do not.

So a dilation gives you a shape that is the same shape at a different size. Mathematicians call that similar, and the multiplier — 2 here — is the scale factor k.

A dilation from a centre, scale factor 2 a dilation stretches every distance from the centre by the same factor O A A' B' B scale factor k = 2 OA' is twice OA every side doubles every angle stays the same similar, not congruent k > 1 enlarges, 0 < k < 1 shrinks, k = 1 changes nothing

What a dilation does

Pick a centre. For every point of the shape, measure its distance from the centre and multiply that distance by k, staying on the same ray.

With the centre at the origin the rule is beautifully simple: (x, y) → (kx, ky).

The image is a scaled copy — every length multiplied by k, every angle exactly as it was.

Angles are untouched

A dilation stretches distances but never bends anything. That is why the image is the same shape, and why similar figures have equal angles.

Reading the scale factor

If the image is bigger, k > 1. If smaller, k is a fraction between 0 and 1. k = 1 leaves everything alone.

To find k from a picture, divide any image length by the matching pre-image length: k = imageoriginal.

Do it with two different pairs of sides. If you get the same k both times, the figures really are similar.

Reading a scale factor: a side of 4 becomes 10 and a side of 2 becomes 5, so k = 2.5 both times divide an image length by the matching original length — new over old 42 105 originalimage × k k =104= 2.5 k =52= 2.5 same k from both pairs,so the triangles are similar 01 = no change23 shrinksenlarges k = 2.5

Which goes on top?

New over old. If a side of 4 becomes 10, then k = 104 = 2.5 — bigger than one, and the picture got bigger. Sanity check it that way every time.

Similar versus congruent

Congruent means identical. Similar means the same shape at possibly a different scale.

Every congruent pair is similar, with k = 1. Not every similar pair is congruent.

In similar figures: angles are equal, and sides are proportional. Those two sentences solve almost every similarity problem.

Congruent triangles match in every side and angle; similar triangles match in angle while their sides are all multiplied by the same k congruent means identical; similar means the same shape at any scale congruentsame shape, same size: k = 1 similarequal angles, sides × 2: k = 2 3456810 every congruent pair is similar (k = 1); not every similar pair is congruent

Finding a missing side

Set up a proportion with matching sides in matching positions, then cross-multiply.

If △ ABC ∼ △ DEF then ABDE = BCEF = ACDF.

Keep the same triangle on top in every fraction. Flipping one of them midway is the classic slip.

Perimeter scales by k, area by k2

Double every side and the perimeter doubles, but the area becomes four times as large. For volume it is k3.

Worked examples

Dilating from the origin

Dilate A(6, -4) by a scale factor of 12, centred at the origin.

  1. Rule: (x, y) → (12x, 12y) Centre at the origin makes it pure multiplication.
  2. 12(6) = 3 and 12(-4) = -2 Halve both coordinates, signs and all.
  3. Sanity check: k < 1, so the point moved closer to the origin. (3, -2) is indeed nearer than (6, -4).

A'(3, -2)

Finding the scale factor

A rectangle 8 by 12 is dilated into a rectangle 20 by 30. Find k.

  1. Use a matching pair: k = 208 = 2.5 New over old.
  2. Check with the other pair: 3012 = 2.5 ✓ Both give the same factor, so it really is a dilation.

k = 2.5

Missing side in similar triangles

△ ABC ∼ △ DEF with AB = 6, DE = 9 and BC = 8. Find EF.

  1. Scale factor: k = DEAB = 96 = 32 Going from △ ABC to △ DEF.
  2. EF = k · BC = 32 × 8 Every side in △ DEF is 32 times its partner.
  3. EF = 12 Check: 128 = 1.5, matching 96.

EF = 12

Words to know

Dilation
A resizing from a centre point by a scale factor k.
Centre of dilation
The one point that does not move.
Scale factor
The multiplier k. k > 1 enlarges, 0 < k < 1 shrinks.
Similar
Same shape, possibly different size. Written △ ABC ∼ △ DEF.

Mistakes to watch for

Dividing old by new. The scale factor is newold, so a side of 4 becoming 10 gives k = 2.5. Get it upside down and you get 0.4, which claims the picture shrank when you can see that it grew.

Adding the scale factor instead of multiplying. A dilation multiplies every distance from the centre. Adding 2 to each side turns a 3-4-5 triangle into 5-6-7, which is not even the same shape any more.

Assuming area scales by k too. Double every side and you get four times the area, because area is a length times a length and both of them doubled. For volume it is k3, for the same reason.

Flipping one fraction in a proportion. ABDE = BCEF keeps the first triangle on top in both. Turn one over and you are solving a different, wrong equation that often still gives a tidy-looking answer.

Thales of Miletus measured the height of the Great Pyramid around 600 BC using nothing but its shadow and his own.

He waited until his shadow was exactly as long as he was tall, then measured the pyramid's shadow. Same scale factor, so that length was the pyramid's height.

It is said the Egyptian priests were astonished. The trick is in this unit: two similar triangles, one of them enormous.

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