HighSchoolQ

📐 Geometry · Similarity and Right Triangles

The Pythagorean Theorem: notes and common mistakes

Study notes

Key ideas

  • a2 + b2 = c2 for a right triangle with legs a and b and hypotenuse c. The hypotenuse is across from the right angle and is the longest side.
  • Hypotenuse unknown: add the squares of the legs, then take the square root. A leg unknown: subtract (big square minus small square), then take the square root.
  • Give exact answers as simplified radicals and decimals with ≈. Keep the exact value, or extra decimals, until the last step, and round once at the end.
  • The converse: if three side lengths satisfy a2 + b2 = c2 with c the longest side, the triangle is a right triangle. If they do not, it is not a right triangle.
  • Pythagorean triples such as 3-4-5, 5-12-13 and 8-15-17, and their multiples, give whole-number sides.
  • The theorem is for right triangles only.

Steps

  1. Draw the triangle, mark the right angle, and find the hypotenuse: the side across from it.
  2. Write a2 + b2 = c2 with the numbers you know and a letter for the unknown side. Keep the hypotenuse alone on its own side.
  3. Solve for the square of the unknown side: add the squares if the hypotenuse is unknown, subtract if a leg is unknown.
  4. Take the positive square root. Simplify the radical, and give a decimal with ≈ if one is asked for.
  5. Check that the hypotenuse is longer than each leg and shorter than the sum of the legs. Give the units.

Formulas

  • Pythagorean theorem: a2 + b2 = c2
  • Hypotenuse: c = √(a2 + b2)
  • Missing leg: b = √(c2 − a2)
  • Converse test: a2 + b2 = c2, with c the longest side, means a right triangle
  • Multiples of a triple: (ka)2 + (kb)2 = (kc)2

Watch out

  • Only the side across from the right angle is the hypotenuse. Do not call a leg the hypotenuse, and do not assume the unknown side is always c.
  • Square first, then add: for legs 3 and 4, 32 + 42 = 25, which is not (3 + 4)2 = 49 and not 3 + 4 = 7.
  • a2 + b2 gives c2, not c. Take the square root.
  • A missing leg means subtract. Add only when the hypotenuse is the unknown.
  • Do not round until the end, and write ≈ for rounded values.
  • Do not use the theorem on a triangle with no right angle. In the converse test, put the longest side in the c spot.

Why it works

  • The squares on the sides of a right triangle have areas a2, b2 and c2. Filling a big square in two ways shows that the two small areas always add up to the big one.
  • It turns lengths that are hard to measure (a diagonal, the height of a ladder, a straight-line shortcut) into lengths you can measure along the right angle.
  • Other lessons build on it: special right triangles and trigonometry come next, and the distance formula on a coordinate grid is the same relationship.

Common mistakes

✗ MistakeThe unknown side is always c. With hypotenuse 20 and one leg 12, the other leg is √(122 + 202) = √544.
✓ InsteadOnly the side across from the right angle is c. Here 20 is the hypotenuse, so a2 + 122 = 202 and a2 = 400 − 144 = 256, which gives a = 16.
✗ MistakeLegs 8 and 14 give a hypotenuse of 8 + 14 = 22 (adding instead of squaring).
✓ InsteadSquare first: 82 + 142 = 64 + 196 = 260, so c = √260 = 2√65 ≈ 16.12. The hypotenuse is always shorter than the sum of the legs.
✗ MistakeLegs 11 and 3 give 112 + 32 = 130, so the hypotenuse is 130.
✓ Instead130 is c2. The hypotenuse is c = √130 ≈ 11.40, only a little longer than the leg 11.
✗ MistakeFor legs 10 and 40, √1700 = 10√17 and √17 ≈ 4.1, so the hypotenuse is 10 × 4.1 = 41.0.
✓ InsteadRounding √17 to 4.1 before multiplying loses accuracy. Keep 10√17, or use √17 ≈ 4.1231, until the end: 10√17 ≈ 41.23.
✗ MistakeA triangle has sides 8 and 9 and no right angle, so its third side is √(82 + 92) = √145.
✓ InsteadThe theorem needs a right angle. Without one, sides 8 and 9 can have many different third sides, depending on the angle between them. So √145 means nothing here.
✗ MistakeTo test sides 24, 32 and 40, I compared 242 + 402 = 2176 with 322 = 1024, so it is not a right triangle.
✓ InsteadThe longest side must be c. Compare 242 + 322 = 576 + 1024 = 1600 with 402 = 1600. They match, so it is a right triangle.
✗ MistakeThe equation is always a2 + b2 = c2 with those exact letters, even if the hypotenuse of my triangle is named a.
✓ InsteadThe letter c is only a name for the hypotenuse. If the hypotenuse is called a, the equation is b2 + c2 = a2. Always keep the hypotenuse alone on its own side.

Teach it back

Explain to a friend (or a pet!) how to find the missing side of a right triangle, and how to test whether three side lengths make a right triangle. Make up your own numbers, and say why the longest side goes by itself in the test.

Explaining it out loud shows what you really know. Did you…

  • I said that the hypotenuse is across from the right angle and is the longest side.
  • I showed adding the squares to find a hypotenuse and subtracting to find a leg, and I took the square root at the end.
  • I gave an exact answer (a radical) and a decimal with ≈, and I rounded only at the end.
  • For the test, I used the longest side as c and compared a2 + b2 with c2.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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