Prove that a triangle with sides x^2 - 1, 2x, and
x^2 + 1 is a right triangle.

Question

Prove that a triangle with sides x^2 - 1, 2x, and 
x^2 + 1 is a right triangle.

anonymous · Answer

What properties do you know for right triangles?

anonymous · Answer

You can use the pythagorean theroem in this problem.

anonymous · Answer

yep! a^2+b^2=c^2
Only problem here which one is a, b and c?

anonymous · Answer

X^2+1 is c because it's the largest.

anonymous · Answer

Maybe

anonymous · Answer

Isn't it the largest though out of the 3 expressions?

anonymous · Answer

what about 2x? Keep in mind that x can be any number

anonymous · Answer

(for this problem this part is not that important)

parthkohli · Answer

No, no... from definition, the Pythagorean triplet is in form $\rm 2x,x^2 - 1,x^2 + 1 $

anonymous · Answer

so which is the hypotenuse?

parthkohli · Answer

The hypotenuse is always $\rm x^2 + 1$ in this case.

anonymous · Answer

@ParthKohli  Thank you

parthkohli · Answer

$$\rm (2x)^2 + (x^2 - 1)^2 = (x^2 +1)^2$$Test.

anonymous · Answer

OK that is right, but you do need to think about it! 2x can be equal to x^2+1 if x=1
(in this problem that makes no sense as x^2-1 would be 0)

parthkohli · Answer

Not all Pythagorean triplets are in form $\rm 2x,x^2 - 1 ,x^2 + 1 $, but all triplets in form $\rm 2x , x^2 - 1, x^2 + 1$ are Pythagorean triplets.

parthkohli · Answer

@Andras That does form a right angled triangle. Remember how $\cos (0)  = 1$?

raden · Answer

just if x>=2

parthkohli · Answer

It works even when $\rm x = 1$.

parthkohli · Answer

In general, always works in case $\rm x \ge 1$.

raden · Answer

1^2-1 = 0

parthkohli · Answer

Practically, it does form a triangle and that is how $\cos(0) = 1$.

parthkohli · Answer

By definition, it is not really a triangle, but is a valid figure.

anonymous · Answer

so this isn't a right triangle.

parthkohli · Answer

Albeit, yes, you can stick with $\rm x\ge 2$.

anonymous · Answer

@ParthKohli  this isn't a right triangle right?

anonymous · Answer

because x =0 and u can't have a negative side.

parthkohli · Answer

For now, it's not a right triangle, but theoretically it is (you'd learn this thing in trigonometry).

raden · Answer

maybe this link can help u (but in indonesian's language) http://androkurniawan.blogspot.com/2011/09/cara-sederhana-mendapatkan-bilangan.html

shubhamsrg · Answer

you may use cosine law..it'll automatically lead you the the fact that angle opposite to longest side is 90 degrees.
longest side, you can easily see is x^2 + 1