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Geometry 9-12 / Week 05 / Shop Day
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Week 05 · Dilations and Similarity

Shop Day

Shop Day: the family scale model
// The scale model and the flashlight
⏱ about 20 min

Shop Day: The Family Scale Model

Saturday the crew clears the kitchen table and lays out cardboard, a tape measure and a ruler.

"A scale model of this room," Comet says. "One tenth. Every length divided by ten."

Wren measures the table. "120 by 80 centimeters. What do you notice about the model?"

Comet's cousin does the division. "12 by 8. And the angles stay square."

Nova projects a tiny room with the furniture glowing. "Would you like a hint? Measure the doorway first. It sets the scale for everything."

"Doorway 210 tall, model 21," the cousin says, marking the cardboard.

"Same shape, one tenth the size," Wren says. "Your turn to build."

  1. A dilation has a center and a scale factor. Each point moves along its ray, k times as far from the center.
  2. Lines through the center stay. Other lines go to parallel lines. Lengths multiply by k, angles stay.
  3. Similar figures are related by rigid motions and dilations.
  4. Similar triangles have equal angles and proportional sides. Two equal angles (AA) are enough.
◇ BUILD A ONE-TENTH MODEL
Pick a room. Measure the floor, the doorway and two pieces of furniture with a tape measure.
Divide every length by ten and write the model sizes in a table beside the real sizes.
A grown-up cuts cardboard rectangles for the floor and the furniture tops. Tape them in place on the model floor.
Check one angle: a square corner in the room should be a square corner in the model. Measure it.
For a grown-up
Do the cutting yourself with scissors. Younger helpers measure, divide and tape.
Let the tape measure wind back slowly, and keep the scissors closed when they are not in use.
If a model piece looks wrong, measure the real piece again together before changing the model.
SHOP DAY CHECK
  • Read the question.
  • Tap your answer.
The family table is 120 cm long. How long is it in the one-tenth model, in centimeters?
The model table is 8 cm wide and the real table is 80 cm wide. What scale factor takes the model to the room?
A square corner in the room is 90°. What is the same corner in the model?
A bookshelf in the room is 180 cm tall. How tall is it in the one-tenth model, in centimeters? Type the number.
WHY THIS EXERCISEOne scale factor for every length is what makes the model similar to the room.
Draw your model floor plan with the real size and the model size written on each piece.
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