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

Thursday

The model plan and the AA criterion
// The scale model and the flashlight
⏱ about 20 min

Thursday: The Model Plan and the AA Criterion

Wren spreads the model plan on the bench. "Every full-size piece in one column, the one-tenth model in the next."

Comet runs down the list. "Flat 240 wide, model 24. Ramp 300 long, model 30. Easy. Divide by ten."

"Now the braces," Wren says, holding up two cardboard triangles, one small, one large. "Are these similar?"

"They look similar. But the big one has no side lengths written on it."

Nova projects both triangles with their angles glowing. "Would you like a hint? Count the matching angles."

"Forty and sixty on both," Comet says. "The third has to be eighty on both. All three angles match."

"Two matching angles were enough," Wren says. "That is the AA criterion. Our designer, work the plan and prove AA."

The crew's model plan

Scale factor 0.1 from the shop floor to the model. Full sizes are the crew's own plan, in centimeters.

PieceFull size (cm)Model (cm)
Flat, width24024
Flat, height36036
Access ramp, length30030
Turntable, diameter20020
Door opening, width909
Door opening, height21021

Similar triangles

Two figures are similar when a sequence of rigid motions and dilations carries one onto the other.

For triangles this means all three pairs of corresponding angles are equal. All three pairs of corresponding sides are in the same ratio.

The crew's two braces: angles 40°, 60°, 80° in both, sides 4, 6, 7 cm and 8, 12, 14 cm. Every side ratio is 2.

Two similar braces side by side, ABC with sides 4, 6, 7 cm and DEF with sides 8, 12, 14 cm.

Proof: the AA criterion

  1. Given: triangles ABC and DEF with angle A = angle D and angle B = angle E.
  2. Angle C = angle F, because both equal 180° minus the other two angles.
  3. Dilate triangle ABC from A by the factor DE divided by AB. Now A′B′ = DE, and the angles are unchanged.
  4. Triangle A′B′C′ and triangle DEF match in angle, side, angle, so they are congruent by ASA.
  5. A congruence is a sequence of rigid motions. A dilation followed by rigid motions carries ABC onto DEF: the triangles are similar.
PUT THE AA PROOF IN ORDER
  • ?Given: two triangles with two pairs of equal angles
  • ?The third angles are equal, since each is 180° minus the other two
  • ?Dilate the first triangle so one side matches the corresponding side of the second
  • ?The dilated triangle and the second triangle are congruent by ASA
  • ?A dilation followed by rigid motions carries one triangle onto the other, so they are similar
WHY THIS EXERCISEAA works because a dilation fixes the size and ASA fixes the rest.
WORK THE PLAN AND THE BRACES
  • Read the question.
  • Tap your answer.
A triangle ABC with angles 40°, 60°, ?Both braces have angles of 40° and 60°. What is the third angle of each?
The small brace's 4 cm side matches the large brace's 8 cm side. The small brace's 7 cm side matches which large side, in centimeters?
The ramp is 300 cm long. How long is it in the one-tenth model, in centimeters?
The model turntable is 20 cm across and the real one is 200 cm. What scale factor takes the model to the real one?
The door opening is 210 cm tall. How tall is it in the one-tenth model, in centimeters? Type the number.
WHY THIS EXERCISEEvery length in the model is the full length times the scale factor.
Two triangles with two pairs of equal angles are similar. The criterion's two letters: ____. Type them.
In the AA proof, after the dilation the two triangles were congruent by which criterion? Type the three letters.
Draw the two braces side by side, label every angle and side, and write the ratio beside each matched pair.

Plan complete, designer. Tomorrow you find similar shapes everywhere and review the week.

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