Converse of Pythagoras Theorem - Proof and Examples

What Is The Converse Of The Pythagoras Theorem?

The converse of the Pythagoras theorem states: in a triangle, if the square of the longest side equals the sum of the squares of the other two sides, then the triangle is a right triangle, with the right angle opposite that longest side. In symbols, if the sides are a, b, c with c the longest, and

a² + b² = c²

then the angle opposite c is 90°.

Proof Of the Converse

The converse is not obvious - knowing a² + b² = c² tells you about lengths, and the claim is about an angle. The proof bridges that gap by building a triangle we know is right-angled and showing the given one must match it.

1. Construct a brand-new right triangle PQR with a right angle at Q, legs PQ = a and QR = b.

2. By the forward Pythagoras theorem applied to PQR, its hypotenuse PR satisfies
   PR² = a² + b².

3. But we were given a² + b² = c², so
   PR² = c² ⟹ PR = c.

4. Now compare triangles ABC and PQR: all three sides match (a = a, b = b, and the third side c = PR). By the SSS (side-side-side) congruence rule, the two triangles are congruent.

5. Congruent triangles have equal corresponding angles, so the angle in ABC opposite c equals the angle in PQR opposite PR - which is the 90° we built. The triangle is right-angled.

The Test Also Sorts Acute And Obtuse Triangles

The converse does more than spot right triangles. Compare a² + b² with c² (where c is the longest side) and you can classify any triangle by its biggest angle:

• If a² + b² = c² → right triangle.
• If a² + b² > c² → acute triangle.
• If a² + b² < c² → obtuse triangle.

Examples of the Converse of the Pythagoras Theorem

Example 1

A triangle has sides 6, 8, and 10. Is it right-angled?
The longest side is 10, so test 6² + 8² against 10²: 6² + 8² = 36 + 64 = 100 = 10².
Final answer: yes, it is a right triangle.

Example 2

A triangle has sides 8, 10, and 6. A student computes 10² and compares it improperly. Is their conclusion correct?
They need to square the two shorter sides and compare against the longest: 6² + 8² = 100 = 10². Final answer: it is a right triangle, as proven.

Example 3

Classify the triangle with sides 6, 8, and 11.
Longest side is 11. Compare 6² + 8² with 11²: 6² + 8² = 36 + 64 = 100 < 121. Final answer: an obtuse triangle.

Example 4

Classify the triangle with sides 4, 5, and 6.
Longest side is 6. Compare 4² + 5² with 6²: 4² + 5² = 16 + 25 = 41 > 36. Final answer: an acute triangle.

Example 5

A triangle has sides 5, 12, and 13. Show it is right-angled and name the hypotenuse.
Longest side is 13. Test: 5² + 12² = 25 + 144 = 169 = 13². Final answer: right-angled, hypotenuse = 13.

Example 6

A triangle has sides 9, 12, and 15. Is it right-angled?
Longest side is 15. Compute: 9² + 12² = 81 + 144 = 225 = 15². Final answer: yes, the corner is a true right angle.

The Mistakes Students Make Most Often On Converse

Mistake 1: Not identifying the longest side first

• Where it slips in: applying the converse improperly when sides are given in random order.

Mistake 2: Confusing the converse with the theorem

• Where it slips in: assuming given angles instead of checking with sides.

Mistake 3: Forgetting the acute and obtuse cases

• Where it slips in: stopping classification at non-right-angled conclusions.

Conclusion

• The converse of the Pythagoras theorem states that if a² + b² = c², the triangle is right-angled.
• It reverses the theorem.
• The proof shows the relationship between angles and sides accurately.