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

Wednesday

Shop Lab: the flashlight projector
// The scale model and the flashlight
⏱ about 20 min

Wednesday: Shop Lab, the Flashlight Projector

Comet switches off the shop lights and clicks on a flashlight. "Lab day. The flashlight is the center of dilation."

Wren holds a cardboard cutout, 10 centimeters tall, 20 centimeters in front of the bulb. Its shadow falls on the wall.

"The wall is 40 from the bulb," Comet reads from the tape. "Shadow height?"

Wren measures the shadow. "Twenty. Exactly double. What do you notice about the distances?"

"Forty is double twenty. The scale factor is the wall distance divided by the cutout distance."

Nova projects the rays from the bulb through the cutout's corners to the wall. "Would you like a hint? Move the wall."

Comet slides the cardboard wall back to 60. The shadow grows to 30. "Factor three," she says. "Our designer, run it."

What you need

  • A flashlight, set on a stack of books so it does not move.
  • A cardboard cutout of a simple shape, about 10 cm tall, cut by a grown-up with scissors. Tape it to a ruler as a stand.
  • A large sheet of cardboard or a plain wall to catch the shadow, a tape measure and your Shop Book.
Safety first
Never shine the flashlight into anyone's eyes. Point it at the wall only.
A grown-up cuts the cardboard with scissors. Keep the dark room clear of cables and stools so nobody trips.
Nothing heavy is lifted alone, and no one climbs to reach the wall.

Run the lab

  1. Set the flashlight and measure 20 cm in front of the bulb. Stand the cutout there and keep it there.
  2. Set the wall 40 cm from the bulb. Measure the shadow's height and write it down.
  3. Move the wall to 60 cm, then 100 cm. Measure the shadow each time.
  4. For each row, divide the wall distance by 20 and divide the shadow height by 10. Compare the two numbers.
  5. Trace the shadow on paper at one distance. Measure its angles and the cutout's angles.

The crew's lab table

The cutout is 10 cm tall, 20 cm from the bulb. These are the crew's own readings, in centimeters.

Wall distanceShadow heightWall ÷ 20Shadow ÷ 10
40 cm20 cm22
60 cm30 cm33
100 cm50 cm55

The two ratio columns match in every row. The scale factor is the distance ratio, and the shadow is the cutout dilated by it.

The traced shadow had the same angles as the cutout, and its edges ran parallel to the cutout's edges. Properties confirmed.

READ THE LAB TABLE
  • Read the question.
  • Tap your answer.
With the wall at 60 cm, the 10 cm cutout threw a 30 cm shadow. What is the scale factor?
With the wall at 100 cm the scale factor is 5. How tall is the shadow of the 10 cm cutout, in centimeters?
The wall is 100 cm from the bulb and the cutout is 20 cm from it. What is the scale factor?
The cutout has a 3 cm wide notch. With scale factor 3, how wide is the notch's shadow, in centimeters? Type the number.
WHY THIS EXERCISEEvery part of the cutout, not just its height, is multiplied by the same factor.
What the lab showsTrue or false?
Moving the wall farther from the bulb makes the shadow larger.?
The shadow's angles are larger than the cutout's angles.?
40 ÷ 20 = 20 ÷ 10?
The flashlight bulb plays the part of the center of dilation.?
WHY THIS EXERCISEThe flashlight shows the theorem physically: rays from one point, lengths multiplied, angles kept.
Try it
Keep the wall still and move the cutout closer to the bulb instead. Predict the shadow height before you measure.
Then move the cutout farther from the bulb. Which distance ratio gives the new scale factor?
Draw the bulb, the cutout and the wall from the side, with rays from the bulb through the cutout's top and bottom.

Bright lab work, designer. Tomorrow the crew plans the whole scale model and proves that two angles are enough.

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