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

Thursday

Similar figures and the missing side
// The flashlight shadow: same shape, new size
⏱ about 20 min

Thursday: Similar Figures and the Missing Side

Raven opens the Blueprint Book to the model clubhouse front wall. "Twelve wide, eight tall, a door and a window."

"The full-size cardboard clubhouse should be five times bigger," Rocket says. "So I multiply everything by five?"

"What do you notice if you only multiply the width?" Raven asks. Rocket sketches it. "A squashed clubhouse. Ugly."

"Every length by five," he says. "Then the big wall and the model wall are the same shape. Similar."

Nova hovers over the two drawings, her light jumping from corner to corner. "Compare the angles," she says.

"All corners still square," Rocket says. "Ninety degrees in both. And every side is five times."

Nova hums. "Equal angles, one ratio for all the sides. That is what similar means."

What similar means

Two figures are similar when one can be moved onto the other by slides, flips, turns and a dilation.

The moves keep the shape. The dilation changes the size. So similar figures have equal matching angles.

Matching sides all share one ratio, the scale factor. Each big side is k times the small side it matches.

Finding a missing side

Find k from one pair of matching sides: k = big ÷ small. Then any missing side is k times its match.

Model door 2 cm wide, full-size door 10 cm wide: k = 5. A model window of 3 cm becomes 3 × 5 = 15 cm.

These are the crew's own Blueprint Table numbers for the model wall and the full-size plan, in centimeters. The scale factor is 5.

PartModel (cm)Full-size plan (cm)
Wall width1260
Wall height840
Door width210
Door height525
Window side315
A grid with a rectangle 6 by 4 and a smaller rectangle inside it near the bottom, the door

The picture shows the model wall at half size on the grid: 6 by 4 with a door 1 by 2.5. The shape is the same.

READ THE BUILD TABLE
  • Read the question.
  • Tap your answer.
The model wall is 12 cm wide and the full-size wall is 60 cm wide. The model is 8 cm tall. How tall is the full-size wall, in cm?
The model door is 2 cm wide and 5 cm tall. The full-size door is 10 cm wide. How tall is it, in cm?
The model window side is 3 cm and the full-size window side is 15 cm. What is the scale factor?
The model wall has perimeter 40 cm. After the dilation by 5, what is the perimeter, in cm?
WHICH SEQUENCE MAKES THEM SIMILAR?
  • Read the question.
  • Tap your answer.
Triangle (1, 1), (3, 1), (1, 2) must land on (6, 2), (10, 2), (6, 4). Which sequence works?
Triangle (1, 1), (3, 1), (1, 2) must land on (-2, 2), (-6, 2), (-2, 4). Which sequence works?
Triangle (1, 1), (3, 1), (1, 2) must land on (3, 0), (9, 0), (3, 3). Which sequence works?
The model wall is 12 cm wide and the full-size wall is 60 cm wide. What is the scale factor? Type the number.
The model window side is 3 cm. How long is the full-size window side, in cm? Type the number.
A corner of the model wall is 90°. How many degrees is the matching corner of the full-size wall?
StatementTrue or false?
Similar figures have equal matching angles.?
In similar figures, every pair of matching sides has the same ratio.?
Multiplying only the width keeps a rectangle similar to the original.?
A dilation followed by a slide still gives a similar figure.?
WHY THIS EXERCISEThe squashed clubhouse shows what goes wrong when sides do not share one ratio.
Draw the model wall 12 by 8 with its door and window. Beside it, draw the same wall dilated by 0.5.

Similar figures share one ratio and all their angles. Tomorrow two angles alone prove two triangles similar.

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