Question

if any triangle the angle be in the ratio 1 : 2 : 3 . prove that the corresponding side are in the ratio 1 : 3^1/2 : 2

Answer 1

Answer:

Let x be the smallest angle.

Since the angles are in the ratio 1 : 2 : 3, the other angles are 2x and 3x.

Since the angles in a triangle add up to 180°, we have

x + 2x + 3x = 6x = 180°  =>  x = 30°.

So the angles are 30°, 60° and 90°.

This is a right angled triangle that is half an equilateral triangle.

Let y be the shortest side, which is then half of a side of the equilateral triangle.  The hypotenuse is a side of the equilateral triangle, so it is 2y.

By Pythagoras' Theorem, the other side is

√ ( (2y)² - y² ) = √ ( 4y² - y² ) = √ (3y²) = √3 y.

So the ratio of the sides is

1 : √3 : 2