Suppose triangle ABC is a right triangle where sina/x+1 = sinb/x^2+1 = sinc/2x and all sides are rational. Find all the right triangles which meet the above conditions. I appreciate any help.

Question

campbell_st · Answer

the information is written in the rule of sines format... so the side lengths are 
x^2 + 1, 2x and x + 1

well this is what I'd do.... if the triangle is right angled then the hypotenuse is x^2 + 1

so the triangle is |dw:1397343320209:dw|

now use pythagoras' theorem 

$$(x +1)^2 + (2x)^2 = (x^2 + 1)^2$$

so solve the equation for x. 

hope it helps

anonymous · Answer

Thank you for the help! If you did the pythagoras' theorem for the equation what answer would you get? The one I am getting doesn't seem right.

campbell_st · Answer

well you end up the a degree 4 trinomial... that gives x = 0 as an answer... 

there is a 2nd solution that's easy to spot when you remove the common factor

anonymous · Answer

Okay. Thank you that helps. I was getting like the squareroot of five