Solve problems involving the converse of the Pythagoras Theorem.

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.

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.