Question 1182225: Can the converse of the Pythagorean theorem determine which triangle with the given three side lengths is a right triangle.
16,21,24
4,9,12
20,21,29
5,12,14
Found 4 solutions by mananth, Edwin McCravy, ikleyn, CPhill:
Answer by mananth(16949)   (Show Source):
You can put this solution on YOUR website!
16,21,24
choose the longest side. If it is a right triangle then 24 will be the hypotenuse.
Let's try
24 ^2 = 576 ----------(1)
16^2+21^ = 697----------(2)
(1) not equal to (2)
so it is not a right triangle
If they are equal then it is a right triangle
Answer by Edwin McCravy(20077)  (Show Source):
You can put this solution on YOUR website! 16,21,24.
162 + 212 = 242
256 + 441 = 576
697 = 576
Nope!
4,9,12
42 + 92 = 122
16 + 81 = 144
97 = 144
Nope!
20,21,29
202 + 212 = 292
400 + 441 = 841
841 = 841
Yep!
5,12,14
52 + 122 = 142
25 + 144 = 196
169 = 196
Nope!
Edwin
Answer by ikleyn(53617)  (Show Source):
You can put this solution on YOUR website! .
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.
(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.
Solved MENTALLY, from the beginning to the end !
Answer by CPhill(2189)  (Show Source):
You can put this solution on YOUR website! 16,21,24
choose the longest side. If it is a right triangle then 24 will be the hypotenuse.
Let's try
24 ^2 = 576 ----------(1)
16^2+21^ = 697----------(2)
(1) not equal to (2)
so it is not a right triangle
If they are equal then it is a right triangle
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