Pythagoras Theorem [Proof and Simple Applications with Converse]

232. Triangle Classification by Sides · How side lengths dictate angle types

The classification of ΔABC by angle C matches the comparison of AB² with AC² + BC².

ABRIGHT-ANGLEDAB = 400AB = 400AC = 388.1AC = 388.1BC = 99BC = 99∠C = 90°∠C = 90°AB² = 160000AB² = 160000AC²+BC² = 160384AC²+BC² = 160384C
You can classify a triangle by comparing the square of its longest side with the sum of the squares of the other two. If c² = a² + b² it is right-angled; if c² < a² + b² it is acute; if c² > a² + b² it is obtuse.

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Selina ICSE: Pythagoras Theorem [Proof and Simple Applications with Converse]

What this lesson covers

Try to break it

Drag C anywhere. When C sits exactly on the dashed circle (Thales' circle on AB), ∠C = 90° and AB² = AC² + BC². Pull C outside the circle and ∠C goes acute with AB² < AC² + BC². Pull C inside the circle and ∠C goes obtuse with AB² > AC² + BC². The circle is the boundary between the three triangle types.

How you build it

Construct triangle ABC with C on the circle that has AB as its diameter.

  • Point tool: mark point A — one end of the diameter.
  • Point tool: mark point B — the other end of the diameter.
  • Segment tool: join A to B — this is the diameter.
  • Midpoint tool: click A then B — M drops at the exact centre of the diameter.
  • Circle tool: click centre M, then click A — the circle on AB as diameter (it passes through B too).
  • Point tool: mark C anywhere on the circle — it snaps onto the curve, so angle ACB is exactly 90°.
  • Segment tool: join A to C.
  • Segment tool: join B to C — triangle ABC is complete, right-angled at C.

The proof, step by step

Prove that comparing AB² with AC² + BC² classifies the triangle by its angle at C.

  • Given: AB is the largest side of ΔABC.
  • Construct ΔABD such that ∠D = 90°, AD = AC, and BD = BC.
  • In right ΔABD, AB² = AD² + BD² (Pythagoras Theorem). Substitute AD = AC and BD = BC to get AB² = AC² + BC².
  • Since ΔABC and ΔABD have equal corresponding sides, they are congruent (SSS). Thus, ∠C = ∠D = 90°.

Worked example

In ΔABC, AB = 13 cm, BC = 12 cm, and AC = 5 cm. Which of the following is true?

AB² = 169, AC² + BC² = 25 + 144 = 169. Since AB² = AC² + BC², ΔABC is right-angled at C by the converse of Pythagoras theorem.

  • ΔABC is right-angled at C — correct
  • ΔABC is obtuse-angled at C
  • ΔABC is acute-angled
  • ΔABC is equilateral
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