Answer:
m+n = 337
Step-by-step explanation:
Lets start from the first triangle, It is given to be as 30-60-90.
The hypotenous of first rectangle be 2x, then other sides are by default x and
[tex]\sqrt{3}(x)[/tex] , using laws of trigonometry.
(sin(30) = 0.5 and cos(30) = [tex]\frac{\sqrt{3}}{2}[/tex])
The perimeter of first triangle is ,
= [tex]2x + x + \sqrt{3}(x) = x(3+\sqrt{3}) = \sqrt{3}x(1+\sqrt{3})[/tex]
Now, for second triangle, the longer leg is 2x, and similarly again,
other 2 sides are [tex]\frac{2x}{\sqrt{3}} and \frac{4x}{\sqrt{3}}[/tex].
Again the perimeter of triangle comes out as,
= [tex]\sqrt{3}(x)(1+\sqrt{3})(\frac{2}{\sqrt{3}})[/tex]
Thus, the repeating pattern is identified. The consecutive perimeters differ by, multiplying by factor [tex]\frac{2}{\sqrt{3}}[/tex]
Thus, we can say that perimeter of 4th triangle is,
= [tex](\frac{2}{\sqrt{3}})^{3}(X)[/tex], where X is the repeating constant.
And of 12th triangle is,
= [tex](\frac{2}{\sqrt{3}})^{11}(X)[/tex],
Evaluating the above ratio, we get,
= [tex]\frac{81}{256}[/tex]
Thus, m =256 and n=81.
Thus, m+n = 256+81 = 337.