# 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.