Pythagoras Theorem
103. The Side-Length Secret · How squares reveal the triangle's true nature
The relationship between the squares of the sides perfectly predicts the type of angle at C.
Triangle classification by side-square relationship: if c² = a² + b², the angle at C is right (90°). If c² < a² + b², the angle is acute (< 90°). If c² > a² + b², the angle is obtuse (> 90°). A purely numerical test for triangle type.
What this lesson covers
Try to break it
Try to position C so that AB² equals BC² + AC². What happens to the angle at C?
How you build it
Construct a 3-4-5 right triangle.
- Mark point A — the right-angle corner.
- Mark point B.
- Draw segment AB — one leg of the triangle.
- Construct the perpendicular to AB at A — this is the second leg, AC.
- Mark point C on the perpendicular line.
- Draw segment BC — the hypotenuse — to complete the triangle.
The proof, step by step
Prove that comparing the squares of the sides reveals the type of angle at C.
- Assume AB² = BC² + AC².
- Construct a right triangle with legs BC and AC.
- By Pythagoras Theorem, Hypotenuse² = BC² + AC².
- Thus, Hypotenuse = AB.
- By SSS congruence, ∠C = 90°.
Worked example
In ΔPQR, PQ = 10 cm, QR = 24 cm, and PR = 26 cm. Which of the following is true?
Check squares: 10² + 24² = 100 + 576 = 676. 26² = 676. Since PQ² + QR² = PR², by the converse of Pythagoras Theorem, ΔPQR is right-angled at Q.
- It is an acute-angled triangle
- It is an obtuse-angled triangle
- It is a right-angled triangle with ∠Q = 90° — correct
- It is an isosceles triangle