Thirty-Six

Unit 6 · GeometryG2 · Triangles · Aerodynamics & Navigation

Similarity & Triangle Reasoning

Set corresponding sides in order and combine similarity with area and perimeter.

  • 30 min
  • Unlocks: Truss Bracing

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

similar triangles
triangles with equal corresponding angles and proportional sides

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Try it first

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Try this before the explanation. Commit to a setup, even if you are unsure.

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The big idea

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The decision that matters

Write corresponding vertices and sides in the same order. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

Write corresponding vertices and sides in the same order. Keep exact values through the setup; round only when the stem asks.
Core relationship
aaprime=bbprime=ccprime\frac{a}{aprime}=\frac{b}{bprime}=\frac{c}{cprime}

Use the formula only after matching every symbol to the diagram or equation.

Algebra or Desmos?

Set up corresponding ratios by geometry before solving numerically. Desmos can solve the proportion once height is matched with height and shadow with shadow.

Explore in Desmos

Solve a shadow proportion

After matching height with height and shadow with shadow, enter the solved proportion and read h=22.5h=22.5 feet from the displayed value.

The calculator is live: change anything and see what happens.

Solve a shadow proportion

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Trap

The main trap

A length scale factor kk becomes k2k^2 for area.

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Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1

Problem

A 6-foot person casts an 8-foot shadow while a tree casts a 30-foot shadow. How tall is the tree?
The sun creates similar triangles.

Which correspondence keeps height over shadow in both ratios?

The tree is 22.522.5 feet tall.

Why are the two shadow triangles similar?

Worked example

Example 2: ACT twist

Problem

Two similar triangles have corresponding sides 5 and 12. If the smaller area is 50, what is the larger area?
Length scale factor is 125\frac{12}{5}.

By what factor does the area change?

The larger area is 50(125)2=28850(\frac{12}{5})^2=288.

Why is the length factor squared?

Faded example · you finish it

Example 3: You finish it

Problem

A triangle with sides 4, 7, 9 is scaled so the shortest side is 10. Find the new longest side.
k=10/4=2.5k=10/4=2.5

Use the scale factor 10/4=2.510/4=2.5 on the original longest side.

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Practice

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.