Answer
28
Work Step by Step
Since all trapezoids add up to 360 degrees, the angles of the base of the trapezoid are:
$ \frac{360 - 240}{2} = 60 ^{\circ}$
This means that the red arc is given by:
$\frac{60}{2} = 30^{\circ}$
This means that triangle MQT is isosceles, so MT is equal to MQ.
We form a 30-60-90 right triangle that divides QTP into two equal triangles. We call h the height of the trapezoid. From this, we obtain:
$h = .5\times 12 \times \frac{\sqrt{3}}{3} \\ h = 2\sqrt{3}$
Thus, we find MQ:
$ MQ = \frac{2}{\sqrt3} \times h \\ MQ = 4$
Since it is isosceles, we obtain that NP equals 4. In addition, we obtained that MT, which is half of MN, is equal to MQ above. Thus. MT equals 8. Therefore:
perimeter = $ 4 +4+8+12 = 28$