Question

The sides of a right triangle form a Gp.THe tangent of the smallest angle is

Answer 1

The sides of a right triangle form a Gp.

The tangent of the smallest angle is 1/ square root of [(1 + root 3)/2]

Sides of triangle = a/r, a, ar where r is greater than 1

The larger side by hypotenuse.

Since the triangle is right angle, use Pythagoras theorem,

(a/r)^2 + a^2 = (ar)^2

1/r^2 + 1 = r^2

Let r^2 = t

1/t + 1 = t

1 + t = t^2

T^2 – t -1 = 0

T = 1+_ root 5 /2

T = 1 + root 5/2

R^2 = 1 + root 5/2

R = root of [1 + root 5/2]

Tan thetha = (a/r)/a = 1/r

Tan thetha = 1/r = 1 / root of [1 + root 5/2]

Rationalize it,

We will obtain root 2 [root of (root 5 -1)]/4

Hence, the answer can be either root of [root 5 -1]/2 or root of [2/root 5 + 1]