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Converse of Pythagoras Theorem

The converse of Pythagoras theorem is used to determine whether or not a triangle is right-angled, given the lengths of its sides. It states that:

“If the squares on one side of the triangle is equal to the sum of the squares on the other two sides, then the angle between the two short sides is a right angle”.

 

Example 1 : Determine whether this triangle is right-angled.

Converse of Pythagoras theorem 1

The converse of Pythagoras theorem is used to determine if a triangle is right-angled.

102  =  100

62  =  36

82  =  64

Hence, 62 + 82  =  36 + 64

=  100   =  102

So, ABC is right-angled at A.

 

Example 2 : Is triangle PQR here right-angled?  Converse of Pythagoras theorem_2

Applying the converse of Pythagoras theorem –

112  =  121

72  =  49

92  =  81

Adding the two shorter sides – 72 + 92  =  49 + 81  =  130.

But this is not equal to 121. So triangle PQR is not right-angled.

 

Example 3 : State whether 13, 16, and 17 is a Pythagorean triad.

To check if three numbers form a Pythagorean triad, we use the converse of Pythagoras theorem.

132  =  169,   162  =  256,   and 172  =  289

132 + 162

=  169 + 256

=  425  <>  289 (172)

So, 13, 16, and 17 do not form a Pythagoras triad.