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Geometry 9-12 / Week 05 / Monday
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Week 05 · Dilations and Similarity

Monday

Shrink the whole set
// The scale model and the flashlight
⏱ about 20 min

Monday: Shrink the Whole Set

Comet sets a shoebox-sized cardboard flat beside the full-size one. "The scale model. Everything one tenth."

Wren holds the model up at arm's length and squints past it at the real flat. "Same shape. What do you notice?"

"It is the same flat, just small. Every length is divided by ten. The angles did not change."

Nova hovers over the chalk grid and projects a small triangle, then a big copy with rays fanning from one corner.

"Would you like a hint? Every point slid out along its ray, twice as far from the center."

"That is a dilation," Wren says. "Center, scale factor, done. Last week was slides, flips and turns. This is the fourth move."

"The move that changes size," Comet says. "Our designer, help us pin down exactly what it does."

Comet, Wren and Nova in the Scene Shop with the small cardboard scale model in front of the full-size painted flats.

What a dilation is

A dilation needs a center O and a scale factor k. It sends each point P to a point P′ on the ray from O through P.

The distance OP′ is k times the distance OP. The center itself does not move.

A factor bigger than 1 enlarges. A factor between 0 and 1 shrinks. A factor of 1 changes nothing.

A triangle at (1, 1), (4, 1), (2, 3) and its image after a dilation from the origin by 2, with rays.

A solved problem to study

Wren dilates the cardboard triangle (1, 1), (4, 1), (2, 3) from the origin with scale factor 2. Here is his work.

Every coordinate is multiplied by 2, because the center is the origin: A (1, 1) goes to A′ (2, 2).

B (4, 1) goes to B′ (8, 2), and C (2, 3) goes to C′ (4, 6).

Check one length: AB is 3 units and A′B′ is 6 units, exactly 2 times as long.

Why does it work? A point at (x, y) is x across and y up from the origin. Doubling its distance along the ray doubles both.

When the center is not the origin, measure from the center instead. From C (1, 1), the point (3, 2) is 2 across and 1 up.

Scale factor 3 makes that 6 across and 3 up, so the image is (7, 4).

WHERE DOES THE POINT LAND?
  • Read the question.
  • Tap your answer.
B at (4, 1) is dilated from the origin with scale factor 2. Where does B′ land?
C at (2, 3) is dilated from the origin with scale factor 2. Where does C′ land?
The point (3, 2) is dilated from center (1, 1) with scale factor 3. Where does it land?
The point (4, 6) is dilated from the origin with scale factor 1/2. Where does it land?
LENGTHS AND FACTORS
  • Read the question.
  • Tap your answer.
AB is 3 units long. After a dilation with scale factor 2, how long is A′B′, in units?
The model flat is 24 cm wide and the real flat is 240 cm wide. What scale factor takes the model to the real flat?
The real flat is 240 cm wide and the model is 24 cm wide. What scale factor takes the real flat to the model?
The transformation with a center and a scale factor is called a ____. Type the word.
The fixed point that every ray starts from is the ____ of the dilation. Type the word.
StatementTrue or false?
The center of a dilation does not move.?
A scale factor of 1/10 makes a figure larger.?
3 × 2 = 6?
A dilation moves every point along a ray from the center.?
A dilation is one of the three rigid motions.?
WHY THIS EXERCISEKeeping dilations separate from rigid motions is what makes "similar" different from "congruent".
Draw the triangle (1, 1), (4, 1), (2, 3) and its image dilated from the origin by 2. Add the three rays.
Try it
On graph paper, mark the origin and draw a small triangle near it. Draw a ray from the origin through each corner.
Double each corner's distance along its ray and join the new corners. Measure one old side and its new side.

Strong start, designer. Tomorrow you test what a dilation does to lines and lengths, and prove it.