Quick Notes

Quick Notes

Mathematics Chapter 13: The Pythagoras Theorem
13.2 The Converse of Pythagoras’ Theorem
 
13.2  The Converse of Pythagoras’ Theorem
 
Determine whether a triangle is a right-angled triangle:
 
  • If \(c^2\lt a^2+b^2\), then the angle opposite to side \(c\) is an acute angle.
  • An acute angle is an angle less than \(90^\circ\).
 

 
  • If \(c^2\gt a^2+b^2\), then the angle opposite to side \(c\) is an obtuse angle.
  • An obtuse angle is an angle more than \(90^\circ\) and less than \(180^\circ\).
 

 

The converse of Pythagoras’ theorem states that:

  • If \(c^2= a^2+b^2\), then the angle opposite to side \(c\) is a right angle.
 

 
Example

Determine whether the following triangle is a right-angled triangle.

The longest side is \(34\text{ cm}\).

So, \(34^2=1\,156\).

Next,

\(\begin{aligned} 18^2+30^2&=324+900 \\\\&=1\,224. \end{aligned}\)

Thus, the triangle is not a right-angled triangle.