Question 1182225
.
Can the converse of the Pythagorean theorem determine which triangle 
with the given three side lengths is a right triangle.
(a) 16,21,24
(b) 4,9,12
(c) 20,21,29
(d) 5,12,14
~~~~~~~~~~~~~~~~~~~~


Yes, it can.


Regarding this concrete problem, the most part of options (a) - (d) can be analyzed MENTALLY,
without making real calculations.


<pre>
(a)  We want to check if 

         16^2 + 21^2 = 24^2.


     In this hypothetical equality, two terms, 16^2 and 24^2, are even integer numbers, while 21^2 
     is odd integer number.  Hence, this equality is not possible - this triangle is not a right triangle.



(b)  We want to check if 

         4^2 + 9^2 = 12^2.


     In this hypothetical equality, two terms, 4^2 and 12^2, are even integer numbers, while 9^2 
     is odd integer number.  Hence, this equality is not possible - this triangle is not a right triangle.



(d)  We want to check if 

         5^2 + 12^2 = 14^2.


     In this hypothetical equality, two terms, 12^2 and 14^2, are even integer numbers, while 5^2 
     is odd integer number.  Hence, this equality is not possible - this triangle is not a right triangle.



(c)  We want to check if 

         20^2 + 21^2 = 29^2.


     In this case, this reasoning with odd-even numbers does not work, so we should check it performing 
     direct explicit computation.

     Left side is  20^2 + 21^2 = 400+ 441 = 841.  Right side is 29^2 = (30-1)^2 = 900 - 2*30 + 1 = 841.

     Both sides are equal - hence, this triangle is a right-angled triangle.
</pre>

Solved MENTALLY, from the beginning to the end !