Packets  /  Pythagorean Theorem
Grades 7-9Geometry

Pythagorean Theorem

The theorem that unlocks right triangles: find hypotenuses, missing legs, recognize Pythagorean triples, test right triangles, measure distance on the coordinate plane, and solve real ladder and diagonal problems.

1The Theorem: a^2 + b^2 = c^2

Key idea: In a RIGHT triangle, a^2 + b^2 = c^2, where c is the hypotenuse — the longest side, opposite the right angle.
  1. Identify the right angle; the side across from it is c, the hypotenuse.
  2. Label the two shorter sides a and b (the legs).
  3. Write a2 + b2 = c2.
  4. Only use this on right triangles — it does not work for other triangles.
Worked example
Legs 3 and 4, find c: 32 + 42 = 9 + 16 = 25, so c = √25 = 5.
Square each leg: 32 = 9, 42 = 16. Add: 9 + 16 = 25. Square root: √25 = 5 (side lengths are positive). The 3-4-5 triangle!
Worked example
Name the hypotenuse: right triangle with legs 5 and 12.
The hypotenuse is the longest side, opposite the right angle. Here 52 + 122 = 25 + 144 = 169, so c = 13 — longer than either leg.
Watch out: c is ALWAYS the hypotenuse (longest side). If your answer for a hypotenuse is shorter than a leg, something went wrong.

2Finding the Hypotenuse

Key idea: Finding the hypotenuse? ADD the squares of the two legs, then take the square root: c = √(a^2 + b^2).
  1. Square each leg.
  2. ADD the two squares: c2 = a2 + b2.
  3. Take the square root of the sum.
  4. Check: c must be the longest side.
Worked example
Legs 6 and 8: 62 + 82 = 36 + 64 = 100, c = √100 = 10.
Square each leg: 62 = 36, 82 = 64. ADD the squares: 36 + 64 = 100. Take the square root: c = √100 = 10. Check: 10 is longer than 6 and 8.
Worked example
Legs 9 and 12: 92 + 122 = 81 + 144 = 225, c = √225 = 15.
81 + 144 = 225. √225 = 15. This is just the 3-4-5 triangle scaled by 3!
Watch out: A common error is taking the square root BEFORE adding (√(a^2) + √(b^2) = a + b). Always add the squares first, then root once.

3Finding a Missing Leg

Key idea: Finding a missing LEG? SUBTRACT the square of the known leg from the square of the hypotenuse, then take the square root.
  1. Write leg2 + (known leg)2 = hyp2.
  2. SUBTRACT the known leg squared from the hypotenuse squared.
  3. Take the square root.
  4. Check: the leg must be shorter than the hypotenuse.
Worked example
Hypotenuse 13, one leg 5: 52 + b2 = 132 → 25 + b2 = 169 → b2 = 169 − 25 = 144 → b = 12.
Set up with c known: 25 + b2 = 169. SUBTRACT the known square: b2 = 169 − 25 = 144. b = √144 = 12. Check: 12 is shorter than 13.
Worked example
Hypotenuse 10, one leg 6: a2 + 62 = 102 → a2 = 100 − 36 = 64 → a = 8.
a2 + 36 = 100. Subtract: a2 = 64. a = √64 = 8. The missing leg is 8.
Watch out: NEVER add when c is already known: c^2 + a^2 would give a leg LONGER than the hypotenuse, which is impossible.

4Pythagorean Triples

Key idea: A Pythagorean triple is three whole numbers a, b, c with a^2 + b^2 = c^2. Memorize the classics: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25).
  1. Take the three numbers; the largest is the candidate c.
  2. Square the two smaller numbers and add them.
  3. Square the largest number.
  4. If they match, it is a Pythagorean triple.
Worked example
Verify 5-12-13: 52 + 122 = 25 + 144 = 169 = 132. It is a triple!
Square all three: 25, 144, 169. Check: 25 + 144 = 169. The sum of the two smaller squares equals the largest — that is the test.
Worked example
Is 8, 15, 17 a triple? 82 + 152 = 64 + 225 = 289 = 172. Yes!
64 + 225 = 289. 172 = 289. It checks, so 8-15-17 is a Pythagorean triple — a right triangle with all integer sides.
Watch out: (6, 8, 10) IS a triple — it is just (3, 4, 5) doubled! Multiples of triples are still triples. But (4, 5, 6) is NOT: 16 + 25 = 41, not 36.

5Converse: Testing Right Triangles

Key idea: The converse of the theorem: if a^2 + b^2 = c^2 for the three side lengths (largest = c), the triangle IS a right triangle.
  1. Order the three sides; the largest is c.
  2. Compute a2 + b2.
  3. Compute c2.
  4. If equal → right triangle. If not → not a right triangle.
Worked example
Sides 9, 12, 15: does 92 + 122 = 152? 81 + 144 = 225 and 152 = 225. Equal → right triangle!
Largest side is 15: 152 = 225. The two smaller: 81 + 144 = 225. They match, so by the converse the triangle IS right-angled.
Worked example
Sides 7, 8, 10: 72 + 82 = 49 + 64 = 113, but 102 = 100. 113 ≠ 100 → NOT a right triangle.
113 does not equal 100, so the converse fails. This triangle is not right-angled.
Watch out: Always square the LARGEST side as c. Using a smaller side as c guarantees a false answer.

6Distance on the Coordinate Plane

Key idea: The distance between two points is the hypotenuse of the right triangle made from their coordinate differences: d = √((x2−x1)^2 + (y2−y1)^2).
  1. Find the x-difference: |x2 − x1| (a leg).
  2. Find the y-difference: |y2 − y1| (a leg).
  3. Square each difference and ADD.
  4. Take the square root: d = √(dx2 + dy2).
Worked example
Distance between (1, 2) and (4, 6): differences 3 and 4 → d = √(32 + 42) = √25 = 5.
dx = 4 − 1 = 3, dy = 6 − 2 = 4. The points form a right triangle with legs 3 and 4. Hypotenuse = √(9 + 16) = √25 = 5.
Worked example
Distance between (−2, 1) and (2, 4): dx = 4, dy = 3 → d = √(16 + 9) = √25 = 5.
2 − (−2) = 4 and 4 − 1 = 3. Again a 3-4-5 right triangle: √(16 + 9) = 5.
Watch out: Watch the signs when subtracting coordinates: 2 − (−2) = 4, not 0. But squaring kills the sign anyway, so |dx| works.

7Word Problems: Ladders and Diagonals

Key idea: Ladders, ramps, screens, and diagonals all hide right triangles: draw the triangle, label the hypotenuse, then add or subtract squares.
  1. Sketch the right triangle in the situation.
  2. Decide: is the missing side the hypotenuse (ADD) or a leg (SUBTRACT)?
  3. Set up a2 + b2 = c2 and solve.
  4. Include units, and sanity-check the size.
Worked example
10-ft ladder, base 6 ft from the wall. How high? 62 + h2 = 102 → h2 = 100 − 36 = 64 → h = 8 ft.
The wall and ground make a right angle: base 6 is a leg, the ladder 10 is the hypotenuse. Subtract: h2 = 64, so h = 8 ft.
Worked example
Rectangle 9 in by 12 in — diagonal? d = √(92 + 122) = √225 = 15 in.
The diagonal splits the rectangle into two right triangles. d2 = 81 + 144 = 225. d = 15 in.
Watch out: The ladder is always the hypotenuse — it leans and is the longest part. And answers like '8' mean nothing without 'feet'.
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60-Second Challenge

How many pythagorean theorem problems can you solve in 60 seconds?