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Question

If the angles of triangle are in the ratio 1:1:4 then ratio of the perimeter of triangle to its largest side

$\sqrt{3} + 2 : \sqrt{2}$
$\sqrt{3} + 2 : \sqrt{3}$
$\sqrt{2} + 2 : \sqrt{3}$
$3 : 2$

hint Hint:

In this question, we have to find the ratio of the perimeter of the triangle to the largest side of the triangle. As we know that the sum of angles of triangle is 180 degrees, so we can find the angles using the ratio. Now, let's assume the length of a=b=y so using the cosine formula we can find the value of c. And later find the ratio of the perimeter (i.e., a +b +c) to largest side (i.e., c).

The correct answer is: $square root of 2 plus 2 colon square root of 3$

$R a t i o space o f space a n g l e s space angle A colon angle B colon angle C space equals space 1 colon 1 colon 4 W e space k n o w space t h a t space s u m space o f space a n g l e s space o f space t h e space t r i a n g l e space a r e space 180 degree. S o comma space angle A plus angle B plus angle C space equals 180 rightwards double arrow space x plus x plus 4 x equals 180 rightwards double arrow 6 x equals space 180 rightwards double arrow x equals 30 angle A equals 30 degree comma space angle B equals 30 degree comma space angle C equals 120 degree space l e t space a equals b equals y C o sin e space f o r m u l a comma space space space space space c squared equals a squared plus b squared minus 2 a b space cos space C rightwards double arrow space c squared space equals y squared plus y squared minus 2 y squared cos space C rightwards double arrow space c squared equals 2 y squared minus 2 y squared cos space C rightwards double arrow space c squared equals 2 y squared left parenthesis 1 minus cos space C right parenthesis rightwards double arrow space c squared equals 2 y squared. space 2 sin squared C over 2 rightwards double arrow space c squared equals open parentheses 2 y space sin space C over 2 close parentheses squared rightwards double arrow c equals 2 y space sin space C over 2 rightwards double arrow c equals 2 y space sin space 60 degree rightwards double arrow c equals 2 y fraction numerator square root of 3 over denominator 2 end fraction rightwards double arrow c equals square root of 3 T h e space r a t i o equals fraction numerator a plus b plus c over denominator c end fraction equals fraction numerator y plus y plus square root of 3 y over denominator square root of 3 y end fraction equals fraction numerator 2 plus square root of 3 over denominator square root of 3 end fraction equals 2 plus square root of 3 colon square root of 3$

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