Angles Ratio

Geometry Level 2

If the angles of a triangle are in the ratio $4:1:1,$ then the ratio of the longest side to the perimeter is $\text{\_\_\_\_\_\_\_\_\_\_}.$

1:6

1 : \big(2 + \sqrt{3}\big)

\sqrt{3} : \big( 2 + \sqrt{3} \big)

2:3

2 solutions

Ankush Gogoi
Sep 17, 2014

The angles are $4x,x,x$ but they must sum to 180 degrees, so the angles are 120 degrees, 30 degrees, and 30 degrees.

By the sine rule,

$\frac{\sin 120^\circ}{c} =\frac{\sin 30^\circ}{b},$ so $c=b\sqrt 3.$ The perimeter is $b+b+ b\sqrt3 = b(2+ \sqrt 3),$ so the ratio of the longest side to the perimeter is $b\sqrt 3 : b(2+\sqrt 3) = \sqrt 3 : (2+\sqrt 3).$

Arpit Dhimanj, the maker of this problem, please take a note of the fact that you can use LaTeX in the options, just write your math term wrapped in

\ ( ................. \ )

The code for square root is \sqrt , so the code \sqrt{3} wrapped in \ ( ... \ ) will give you the output $\sqrt{3}$

Aditya Raut - 6 years, 8 months ago

Christian Daang
Sep 18, 2014

the angles will be 30, 30, 120

so, half the 120 and the longest side will be halved also..

so, draw a imaginary line to avoid confusion...

so, if the 2 sides is s,

then, the longest side will be: 2(sV3)/2 or sV3

so, the Perimeter will be: s+s+sV3 or 2s+sV3

So the ratio will be: sV3 : 2s+sV3

= (V3) : (2+V3)

Angles Ratio

2 solutions

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