Sydney Opera House

Geometry Level 2

Solve for $a.$

9

9 \sqrt{3}

18

6 \sqrt{3}

1 solution

Eli Ross Staff
Mar 8, 2016

Let's look at one iteration. In the smaller triangle, with side opposite the $30^\circ$ angle equal to $x,$ the side opposite the $60^\circ$ angle is equal to $x\sqrt{3}.$ Thus, the hypotenuse of the larger triangle is equal to $2x\sqrt{3},$ so its shortest leg is equal to $\frac{2x\sqrt{3}}{2} = x\sqrt{3}.$ In other words, the scale factor between each triangle is $\sqrt{3},$ so the answer is $\sqrt{3} \times \left(\sqrt{3}\right)^3 = \left(\sqrt{3}\right)^4 = 3^2 = 9.$

Easy to see that he area of each smaller triangle will fit 3 times in the next. so the smallest triangle will fit 3^3 times in the largest. So the side of the largest traingle will be sqrt3 x sqrt(3^3) = 9.

Fedde Reeskamp - 5 years, 3 months ago

Sydney Opera House

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