Theorem of Pythagoras (converse)

The converse of the theorem says that if, \({a^2} = {b^2} + {c^2}\) then you have a right-angled triangle and furthermore, the right angle is directly opposite \(a\) (the hypotenuse).

We can therefore use the converse to check whether a triangle is right-angled or not.

Example

A triangle has sides of 5cm and 12cm, with a longest side of 13cm.

Is the triangle right-angled? Explain your answer.

Answer

If the triangle is right-angled then the statement \({13^2} = {5^{^2}} + {12^2}\) will be true.

Longest side (hypotenuse):

\({13^2} = 169\)

Short sides:

\({5^2} + {12^2}\)

\(25 + 144 = 169\)

Therefore, by the converse of the theorem of Pythagoras, the triangle is right-angled since the longest side squared is equal to the square of the other two sides added together.

Now try the example question below.

Question

A triangle has sides of 8cm and 9cm, with a longest side of 12cm.

Is the triangle right-angled? Explain your answer.

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Applying Pythagoras TheoremTheorem of Pythagoras (converse)