The measures of the angles of $△ Q R S$ are $m ∠ Q = 2 x + 4, m ∠ R = 4 x - 12,$ and $m ∠ S = 3 x + 8.$ $Q R = y + 9, R S = 2 y - 7$ , and $Q S = 3 y - 13.$ The perimeter of $△ Q R S$ is:

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Solution

The correct option is A 55
It is given that the measures of the angles of triangle $Q R S$ are $m ∠ Q = 2 x + 4$ , $m ∠ R = 4 x - 12$ and $m ∠ S = 3 x + 8$ and also we know that the sum of the measures of the angles of a triangle is $180$ .
Therefore, $2 x + 4 + 4 x - 12 + 3 x + 8 = 180$
$\Rightarrow 9 x = 180$
$\Rightarrow x = 20$
Substitute the value of $x$ in the measures of the angles that is:
$m ∠ Q = 2 (20) + 4 = 44$
$\Rightarrow m ∠ R = 4 (20) - 12 = 68$
$\Rightarrow m ∠ S = 3 (20) + 8 = 68$
Therefore, $△ Q R S$ is isosceles and $Q R = Q S$ which means
$y + 9 = 3 y - 13$
$\Rightarrow 3 y - y = 13 + 9$
$\Rightarrow 2 y = 22$
$\Rightarrow y = 11$
Now consider the sum of the sides that is:
$Q R + R S + Q S = y + 9 + 2 y - 7 + 3 y - 13 = 6 y - 11$
Hence, the perimeter of triangle $Q R S$ is:
$6 y - 11 = 6 (11) - 11 = 66 - 11 = 55$

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