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Understand Similarity Through Dilations

Students perform dilations on figures by multiplying each coordinate by a scale factor from a center point, and recognize that the resulting figure is similar (same shape, proportional side lengths, equal angles). They describe a similarity transformation as a dilation possibly composed with rigid motions.

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Understand Similarity Through Dilations

Grade 8 · Geometry: Transformations · question 1 of 3

0 of 20

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ABA′B′−8−6−4−22468−8−6−4−224680

What scale factor takes segment AB to segment A′B′ from the origin?

Compare the figure with its image.

How it is taught

0:00 of 4:15

A dilation from the origin multiplies each coordinate by the scale factor.

The new figure is similar to the old one.

BCD−8−6−4−22468−8−6−4−224680

Read it: Dilations from the origin

Segment TX is dilated by a scale factor of 3 from the origin. Every coordinate is multiplied by 3, so T(1, 2) lands at T′(3, 6) and the image is 3 times as long.

  1. Multiply each coordinate by the scale factor.
  2. To find the factor, divide an image coordinate by the matching original one.
  3. Above 1, the image grows; between 0 and 1, the image shrinks.

A dilation from the origin is (x, y) → (kx, ky) for the scale factor k.

The image has the same shape and angles, with every length times k, so it is similar to the figure.

The origin stays put; every other point moves along the line from the origin.

A dilation multiplies every coordinate by the scale factor.

Adding the factor: sending T(1, 2) to (4, 5).

Scale multiplies every distance from the center: 3 × 1 = 3 and 3 × 2 = 6, so T′ is (3, 6).

Point D(−3, −2) is dilated by a scale factor of 4 from the origin. What is the y-coordinate of D′? — −8. Multiply y by the factor: 4 × (−2) = −8.

Dilate △ABC by scale factor 2 from the origin

Today we'll learn dilations — stretching a shape bigger or smaller from a center point.

Here's triangle ABC. A is at one-one, B at three-one, and C at one-three.

Grade 8 · Same shape, sides twice as long

We'll dilate from the origin — that's our center of dilation.

To dilate by a scale factor of two, we multiply each coordinate by two.

A at one-one becomes A prime at two-two.

B at three-one becomes B prime at six-two.

C at one-three becomes C prime at two-six.

Connect the new vertices. The image triangle has the same shape — just bigger!

Every side is exactly twice as long, and every angle stays the same.

The two triangles are similar — that's exactly what a dilation does!

Now you try. Find the x-coordinate of A prime. Type that coordinate.

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
Climbing now
Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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