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Maths-

General

Easy

Question

Two sides of triangle are given by equation and angle between
the sides is $pi divided by 3.$ Then perimeter of the triangle is

$5 + \sqrt 2$
$5 + \sqrt 3$
$5 + \sqrt 5$
$\sqrt{5 + \sqrt 7}$

hint Hint:

We will use the properties of quadratic equation. Sum of roots and product of roots. Then we will apply the cosine rule to find the third side and then we will calculate the perimeter

The correct answer is: $square root of 5 plus √ 7 end root$

In this question we are given sides of triangle as $x squared minus 5 x minus 6 equals 0$ and the angle between the is $pi over 3$ .
Step1: Finding the sides.
Let $a$ , $b$ be the sides with are roots of the given equation $x squared minus 5 x minus 6 equals 0$ .
We know that
Sum of roots: $a plus b equals negative left parenthesis negative 5 right parenthesis$
Product of roots $a b space equals space minus 6$
Step2: Using the Cosine Rule.
We are given angle between side $a$ and side $b$ . That is $angle C equals 60 degree$
$cos open parentheses C close parentheses equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction$
=> $a squared plus b squared equals left parenthesis a plus b right parenthesis squared minus 2 a b$
=> $cos open parentheses 60 degree close parentheses equals fraction numerator 5 squared minus 2 left parenthesis negative 6 right parenthesis minus c squared over denominator 2 cross times left parenthesis negative 6 right parenthesis end fraction$
=> $1 half equals fraction numerator 5 squared minus 2 left parenthesis negative 6 right parenthesis minus c squared over denominator 2 cross times left parenthesis negative 6 right parenthesis end fraction$
=> $1 equals fraction numerator 25 plus 12 minus c squared over denominator negative 6 end fraction$
=> $c squared equals 25 plus 12 plus 6$
=> $c equals square root of 43$
So, the perimeter of the triangle is $a plus b plus c equals space 5 plus square root of 43$

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