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Question

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


A

2+2:3

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B

3:2

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C

3+2:2

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D

3+2:3

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Solution

The correct option is D

3+2:3


Explanation for the correct option.

Step 1: Find the angles

Given that, the angles of a triangle are in the ratio1:1:4.

Let A,B,C be the angles of the triangle.
&A=x,B=x,C=4x

Now using the angle sum property of triangle we have:
A+B+C=180°x+x+4x=180°6x=180°x=30°
B=A=30°andC=4x=120°

Step 2: Find the length of the sides

Now using sine rule, asinA=bsinB=csinC
asin30°=bsin30°=csin120°a12=b12=c32b=aandc=3a

Step 3: Ratio of the perimeter of the triangle to its largest side

The perimeter of the triangle is given by the sum of the length of all sides.

P=a+b+c=a+a+3a=(2+3)a

Largest side=3a

So, the ratio of the perimeter of the triangle to its largest side is:

P erimeterLargest Side=(2+3)a3a=2+33

Hence, option D is correct.


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