Right triangle where sin(alpha)/(x+1)=sin(beta)/(x^2+1)=...

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messa

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Is there a better or faster way to do this problem than guess and check?

Suppose triangle ABC is a right triangle where sin alpha/x+1=sin beta/x^2+1=sin gamma/2x. All sides are rational. Find all right triangles which meet the conditions.
 
Right Triangle
Law of Sines
Pythagorean Theorem

That is all you need.
 
Taking the previous tutor's reply to heart:

I would start by graphing x + 1, x^2 + 1, and 2x on one graph, finding their intersection point(s) and relative heights.

Then review the Law of Sines, and think about how the value of the hypotenuse relates to the values of the lengths of the two legs of a right triangle.

Then consider x-values in various ranges, given by the graph in the first step. For instance, you know that x has to be positive (since otherwise 2x would be negative or zero). What if x = 1? What if x < 1? Or x > 1?

Apply the Pythagorean Theorem, and see where it leads.

If you get stuck, please reply showing all of your work and reasoning. Thank you.

Eliz.
 
I don't even understand the graph of the triangle. How could something be a triangle if there is an x^2 on side b. That's a parabola and not a straight line. Where is the triangle formed?
 
You are somewhat missing the point.

You have:
x+1
x^2 + 1
2x

These are the sides of a RIGHT triangle. The problem is, you can't tell which is the Hypotenuse. This is a matter of investigation.

First off, since it is a triangle, x > 0.

Graphing the three:
y = x+1
y = x^2 + 1
y = 2x
shows that x + 1 is the greatest for x < 1.

Solve (x^2 + 1)^2 + (2x)^2 = (x+1)^2 for x. If you get something 0 < x < 1, you have a solution.

Further, the graph shows that x^2 + 1 is the greatest for x > 1.

Solve (x^2 + 1)^2 = (2x)^2 + (x+1)^2 for x. If you get something x > 1, you have a solution.
 
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