# A right triangle abc with sinA/x+1=sinB/x^2+1=siny/2x..How could i find all the right triangles which meet the conditions?all saides are rational A=alpha B=beta y=gamma

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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Since the triangle ABC is a right triangle, then one of it's angles is 90 degrees.

We'll put A = 90 degrees.

That means that B+C = 90 degrees.

We know that sin 90 = sin A = 1.

We'll re-write the condition given by enunciation:

sin A/(x+1) = sin B/(x^2 + 1) = sin C/2x

We'll substitute sin A by 1:

1/(x+1) = sin B/(x^2 + 1) = sin C/2x

We'll take the 2nd and the 3rd ratios and we'll use the following property:

a/b = c/d

(a+c)/(b+d) = c/d

sin B/(x^2 + 1) = sin C/2x

(sin B + sin C)/(x^2 + 2x + 1) = sin C/2x

(sin B + sin C)/(x + 1)^2 = sin C/2x

1/(x+1) = sin C/2x

sin C = 2x/(x+1)

(sin B + sin C)/(x + 1)^2 = 1/(x+1)

(sin B + sin C)/(x + 1) = 1

sin B + sin C = x+1

From all the identities above, we'll get 2x = x+1.

2x - x = 1

x = 1

The right angle triangles that satisfy the given constraint have x = 1.