Q.31. Application of geometry/trigonometry.

Question 31.

The figure marked "Question" shows a right triangle with legs of lengths 3 and 1 and containing four lines of lengths "x". Find the value of "x".

 

Answer 31.

This challenging question was posted on Yahoo Answers by Dragan K and answered by Rozeta53 and Frst Grade Rocks! Ω.

They all are my valuable contacts and contributors on YA and I am thankful to them for allowing me to put this excellent question and its elegant solution on my blog.

The figure marked "Answer" shows different angles as multiples of θ using basic geometry which leads to the equations,

x = 1/sin4θ    ...   (1)   and tanθ = 1/3   ...   (2).

tan2θ = 2tanθ/(1-tan^2 θ) = 2*(1/3)/[1 - (1/3)^2] = 3/4 and

sin4θ = 2tan2θ/(1+tan^2 2θ) = 2*(3/4)/[1 + (3/4)^2] = 24/25.

Plugging this value of sin4θ in equation (1),

x = 1/(24/25) = 25/24.
Comment posted by Rozeta53 after her answer was selected as the Best Answer is
"Yes, if the shorter leg is a and the longer one is b, then
x = (a² + b²)² / [4b(b² - a²)]".
 

Link to YA!