Thirty-Six

Unit 6 · GeometryG2 · Triangles · Aerodynamics & Navigation

Right Triangles

Use the Pythagorean theorem and special right-triangle ratios efficiently.

  • 30 min
  • Unlocks: Truss Bracing
  • Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

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Key terms

hypotenuse
the side opposite the right angle

Step 2 of 7Try it firstDone

Try it first

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

Step 3 of 7LearnDone

The big idea

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The decision that matters

Label the hypotenuse before substituting into a2+b2=c2a^2+b^2=c^2. 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

Label the hypotenuse before substituting into a2+b2=c2a^2+b^2=c^2. Keep exact values through the setup; round only when the stem asks.
Core relationship
a2+b2=c2a^2+b^2=c^2

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

Algebra or Desmos?

Recognize common triples before reaching for a calculator. For less familiar side lengths, Desmos can evaluate the square-root expression after the hypotenuse and legs have been identified.

Explore in Desmos

Check two Pythagorean triples

Type the Pythagorean expressions; algebra and known triples are faster when visible.

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

Check two Pythagorean triples

Degree mode

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

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Trap

The main trap

Do not add side lengths before squaring them.

Step 4 of 7Worked examplesDone

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 right triangle has legs 9 and 12. Find the hypotenuse.
c2=92+122=225c^2=9^2+12^2=225.
92+122=c29^2+12^2=c^2, so c=15c=15.

Which side is represented by cc?

The triangle is a 3-4-5 triple scaled by 3.

Why is c=15c=15 rather than −15-15?

Worked example

Example 2: ACT twist

Problem

A ladder reaches 24 feet up a wall with its foot 7 feet from the wall. How long is the ladder?
The ladder is the hypotenuse.

Which side of the right triangle is the unknown hypotenuse?

L=25L=25 feet.

Why is the ladder the hypotenuse?

Faded example · you finish it

Example 3: You finish it

Problem

A 4545-4545-9090 triangle has hypotenuse 10210\sqrt2. Find a leg.
h=s2h=s\sqrt2

Complete the calculation.

Desmos way

Evaluate the two hypotenuse expressions

Type the Pythagorean expressions; algebra and known triples are faster when visible.

Evaluate the two hypotenuse expressions

Degree mode

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

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Step 5 of 7PracticeDone

Practice

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Step 6 of 7Spot the mistakeDone

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.