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Geometry 9-12 / Week 05 / Tuesday
2/6
Week 05 · Dilations and Similarity

Tuesday

What a dilation keeps and changes
// The scale model and the flashlight
⏱ about 20 min

Tuesday: What a Dilation Keeps and Changes

Wren tapes a straightedge along side AB of the triangle on the chalk grid, then along A′B′. "Look at the two edges."

"Parallel," Comet says. "Same slant, just farther out. What about a line through the center?"

She lays the straightedge from the origin through A and A′. "It is the same line. A′ is just farther along it."

Nova projects the two triangles with every side measured. "Would you like a hint? Divide each new length by the old one."

"Two, two, two," Comet reads. "Every side doubled. And the angles?"

"Match the corners," Wren says. She lays the small triangle on the big one, corner on corner. They fit.

"So a dilation keeps angles, keeps parallel lines, and multiplies lengths," Comet says. "Our designer, check our numbers."

Three properties to verify

  • A line through the center of dilation goes to itself. Points on it slide along it.
  • A line not through the center goes to a parallel line.
  • Every length is multiplied by the scale factor. Angles stay the same.

The crew's check on the chalk grid

Scale factor 2 from the origin. Every number below was read off the grid or computed from the coordinates.

SideBeforeLengthAfterLengthRatio
AB(1, 1) to (4, 1)3(2, 2) to (8, 2)62
AC(1, 1) to (2, 3)2.24(2, 2) to (4, 6)4.472
BC(4, 1) to (2, 3)2.83(8, 2) to (4, 6)5.662

AB runs along y = 1 and A′B′ along y = 2. Both are horizontal: parallel.

The line from the origin through A (1, 1) is y = x. A′ is (2, 2), still on y = x. The line kept its place.

The ratios in the last column are the lengths after divided by the lengths before. The decimals round, the whole numbers do not.

Why the lengths multiply

  1. Take two points P and Q and the center O. Their images are P′ and Q′ on the same rays, k times as far out.
  2. Triangles OPQ and OP′Q′ share the angle at O, and OP′ is k times OP, OQ′ is k times OQ.
  3. Two sides in the same ratio around a shared angle make the triangles similar (SAS for similarity).
  4. So P′Q′ is k times PQ, and the angle at P′ equals the angle at P. That is why lines go to parallel lines.

Tomorrow's lab checks the same facts with a flashlight instead of a grid. Both methods agree, which is how you know the theorem is true.

PARALLEL OR THE SAME LINE?
  • Read the question.
  • Tap your answer.
Line 1 is AC, through (1, 1) and (2, 3). Line 2 is A′C′, through (2, 2) and (4, 6). Parallel, perpendicular or neither?
Line 1 is BC, through (4, 1) and (2, 3). Line 2 is B′C′, through (8, 2) and (4, 6). Parallel, perpendicular or neither?
The line from the origin through A (1, 1) is dilated from the origin by 2. What happens to the line?
MULTIPLY THE LENGTHS
  • Read the question.
  • Tap your answer.
AC is about 2.24 units. After the dilation by 2, about how long is A′C′, in units?
The door opening is 90 cm wide. In the one-tenth model, how wide is it, in centimeters?
The model flat is 36 cm tall. The real flat is the model dilated by 10. How tall is it, in centimeters?
A 6 cm side of a cardboard shape becomes 9 cm. What was the scale factor?
StatementTrue or false?
A dilation with scale factor 2 doubles every angle.?
A line through the center of dilation is carried onto itself.?
A line not through the center is carried to a parallel line.?
3 × 2 = 6?
A scale factor of 1/2 makes every length half as long.?
WHY THIS EXERCISEThese three properties are the whole reason similar figures look alike.
A dilation sends a line not through the center to a ____ line. Type the word.
A dilation multiplies every length by the scale factor but leaves every ____ unchanged. Type the word.
Try it
On graph paper, draw a small quadrilateral and dilate it from the origin by 3.
Pick one side and its image. Check with a ruler that they are parallel and that the image is three times as long.

Careful checking, designer. Tomorrow is Shop Lab: the flashlight becomes the center of dilation.

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