1. What is the Converse of the Pythagoras Theorem?
The converse of the Pythagoras Theorem tells us how to check whether a given triangle is a right-angled triangle.
It states that:
If in a triangle, the square of the longest side equals the sum of the squares of the other two sides, then the triangle is a right-angled triangle.
2. Formal Statement
If a triangle has sides \(a\), \(b\), and \(c\) such that:
\(c^2 = a^2 + b^2\)
where \(c\) is the longest side, then the triangle is right-angled at the angle opposite side \(c\).
3. Understanding the Idea
The Pythagoras Theorem works only for right triangles. Its converse helps us identify if a triangle is right-angled without measuring any angle directly.
We simply check the relation:
\( \text{(Longest side)}^2 = \text{(Side 1)}^2 + \text{(Side 2)}^2 \)
4. Example
Given side lengths: \(6\), \(8\), \(10\).
Check:
\(10^2 = 6^2 + 8^2\)
\(100 = 36 + 64\)
\(100 = 100\)
Since the equality holds, the triangle is a right-angled triangle with the right angle opposite the side of length 10.
5. Another Example
Given sides: \(5\), \(7\), \(9\).
Check:
\(9^2 = 5^2 + 7^2\)
\(81 = 25 + 49\)
\(81 ≠ 74\)
The relation does not hold, so the triangle is not right-angled.
6. Why the Converse is Useful
The converse of Pythagoras is used to:
- Determine whether a triangle is right-angled,
- Verify right triangles in geometric proofs,
- Check triangle types when angles are not given,
- Solve coordinate geometry problems involving distances.