Pythagoras Theorem
101. The Pythagorean Promise · hypotenuse squared equals the sum of the legs squared
AB²+AC² always equals BC².
The Pythagoras Theorem: in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: AB² + AC² = BC². Drag B or C; the equation always holds when the angle at A is exactly 90°.
What this lesson covers
Try to break it
Try to break it — but AB²+AC² always equals BC², no matter how you drag. That's the Pythagorean theorem!
How you build it
Construct the right triangle, then draw a square on each of its three sides.
- Mark point A — the right-angle corner.
- Draw a horizontal line through A — this is leg AB.
- Construct a line perpendicular to it at A — this is leg AC.
- Mark point B on the horizontal line.
- Mark point C on the perpendicular line.
- Draw the hypotenuse from B to C.
- Draw a square on leg AB.
- Draw a square on leg AC.
- Draw a square on the hypotenuse BC.
The proof, step by step
Prove that AB² + AC² = BC² in a right triangle.
- In right triangle ABC, draw AD perpendicular to BC, meeting BC at D.
- Triangles ABD and ABC are similar (∠ABD = ∠ABC, ∠ADB = ∠BAC = 90°).
- From similarity, AB/BC = BD/AB, so AB² = BD × BC.
- Similarly, ΔACD ~ ΔABC, giving AC² = CD × BC.
- Adding: AB² + AC² = BD×BC + CD×BC = (BD+CD)×BC = BC². Hence, BC² = AB² + AC².
Worked example
In a right-angled triangle ABC, angle A = 90°. If AB = 6 cm and AC = 8 cm, what is the length of the hypotenuse BC?
Using Pythagoras theorem: BC² = AB² + AC² = 6² + 8² = 36 + 64 = 100. Therefore, BC = √100 = 10 cm.
- 10 cm — correct
- 12 cm
- 14 cm
- 9 cm