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Geometry and Measurement 6-8 / Week 10 / Tuesday
2/6
Week 10 · Dilations and Similar Figures

Tuesday

The coordinate rule for a dilation
// The flashlight shadow: same shape, new size
⏱ about 20 min

Tuesday: The Coordinate Rule

Raven tapes a fresh grid to the Blueprint Table. "Rocket, dilate this triangle by 3 without the flashlight."

Rocket frowns at point B at (2, 1). "No bulb, no shadow. How do I know where B lands?"

"What do you notice about A yesterday?" Raven asks. "A was (1, 1). With k = 2, A′ was (2, 2)."

"Both coordinates doubled," Rocket says slowly. "So with 3, every coordinate triples. B′ is (6, 3)!"

Nova hovers over the grid, her light tracing a line from the origin through B to B′. "Same line," she says.

"Three times as far from the origin, along the same line," Raven says. She plots the whole image.

Nova hums. "Now try a scale factor smaller than 1. Predict before you plot."

Multiply every coordinate by k

A dilation from the origin by scale factor k has the rule (x, y) to (kx, ky). Both coordinates are multiplied by k.

Each image point sits on the line from the origin through the original point, k times as far out.

For k = 3: (2, 1) to (6, 3). For k = 0.5: (4, 6) to (2, 3). For k = 1: nothing moves.

Pointk = 2k = 3k = 0.5
(1, 1)(2, 2)(3, 3)(0.5, 0.5)
(2, 1)(4, 2)(6, 3)(1, 0.5)
(4, 6)(8, 12)(12, 18)(2, 3)
(-2, 4)(-4, 8)(-6, 12)(-1, 2)
  1. Write the coordinates of every corner of the original figure.
  2. Multiply each x by k and each y by k.
  3. Plot the new corners and connect them in the same order.
  4. Check: each new side should be k times its old side.
A grid with a gold triangle and its blue image half as far from the origin on every side

A scale factor of 0.5 pulls every corner halfway back toward the origin. The image is half the size.

WHICH RULE?
  • Read the question.
  • Tap your answer.
Which rule gives the new coordinates after a dilation from the origin by 2?
Which rule gives the new coordinates after a dilation from the origin by 3?
Which rule gives the new coordinates after a dilation from the origin by 0.5?
Which rule is not a dilation at all, but a flip across the y-axis?
WHERE DOES IT LAND?
  • Read the question.
  • Tap your answer.
Point (3, 2) is dilated from the origin by a scale factor of 3. Where does it land?
Point (-1, 4) is dilated from the origin by a scale factor of 2. Where does it land?
Point (6, -2) is dilated from the origin by a scale factor of 0.5. Where does it land?
Point (2, 5) is dilated from the origin by a scale factor of 4. Where does it land?
SIDES AFTER A DILATION
  • Read the question.
  • Tap your answer.
A side is 5 cm long. After a dilation by 3, how long is it, in cm?
A side is 8 cm long. After a dilation by 0.5, how long is it, in cm?
A side is 2.5 cm long. After a dilation by 2, how long is it, in cm?
A side 3 units long becomes 12 units long. What scale factor was used?
Triangle ABC is dilated by 3 from the origin. B is at (2, 1). What is the x-coordinate of B′? Type the number.
WHY THIS EXERCISEThe rule (kx, ky) does the whole dilation one coordinate at a time.
Draw triangle (2, 1), (4, 1), (2, 3). Draw its image for k = 2 and its image for k = 0.5 on the same grid.

The rule (kx, ky) dilates any figure. Tomorrow is Build Lab: measure how the flashlight sets the scale factor.

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