How is it mandatory for the r^2 to even? Say the two sides are 3 and 5 - the third side can be anywhere between 2 and 8 - if its 7 then C doesnt hold true.
1 Like
This is a beautiful problem, makes you think of all your concepts, so I think it is a good exercise to start an elimination process. Let me share with you how I did it. Suppose p and q are the two legs of the right triangle, and since they are primes greater than 2 and p\leq q, what about p=q so that we have a right isosceles triangle? Since the question asks about what must be true, we know that the product of the legs will always be an odd number, so when taking the area, we would not get an integer in that case. Thus, we discard B.
Let us continue with the example of a right isosceles triangle, and let p,q be the legs of such triangle. Suppose p=q=5. We know that the hypothenuse is 5\sqrt{2}, so clearly the perimeter of such triangle is not necessarily an integer number. Thus, we eliminate E. Also, by the same example notice that r, the hypothenuse, is not necessarily an integer, so we can also discard A.
Now suppose that we have a right isosceles triangle where p=q=3 and p,q are the legs of such triangle. What is the value of the side r i.e. the hypothenuse? It is 3\sqrt{2}\approx 4.2<5. Thus, we can definitely discard D, and so by our process of elimination, C would be the correct answer.
Now, your question says that if two sides are 3 and 5, the third side can be anywere between 2 and 8, which is correct. However, recall that we have a right triangle, so actually the third side will take a very particular value, because we need to make use of the Pythagorean Theorem. Suppose p=3 is one leg and q=5 is the hypothenuse. By the Pythagorean Theorem (or if you already know your Pythagorean triplets), we know that r=4, so cleary r^2 is even. Now, suppose that p=3 and q=5 are our two legs, so let’s find the hypothenuse by 3^2+5^2=r^2, so r^2=34 which is indeed even.
Thanks for replying! I had such a brain fart. I didnt even notice the question stem mentions its a right triangle.