## anonymous 5 years ago A square with sides of length s is inscribed in an equilateral triangle with sides of length t. Find the exact ratio of the area of the equilateral triangle to the area of the square.

1. watchmath

$$\frac{6}{2+\sqrt{3}}$$

2. anonymous

how did you get that?

3. anonymous

cow, remember me? i need helped, wit hthe problem you helped me yesterday

4. dumbcow

yeah

5. anonymous

hey dumbcow lol can u help me lout with my question

6. anonymous
7. dumbcow

Area of square is s^2 Area of equilateral triangle is $\frac{\sqrt{3}}{4} t ^{2}$ to get ratio we need t in terms of s if we use the right triangle to left of square with side opposite of 60 degree angle the edge of the square of length s and adjacent side length (t-s)/2 tan 60 = opp/adj = 2s/(t-s) = sqrt(3) solve for t $t = \frac{2+\sqrt{3}}{\sqrt{3}} s$ substitute this into Area equation $A = \frac{\sqrt{3}}{4}*\frac{(7+4\sqrt{3})}{3} s ^{2}$ $A = \frac{12 + 7\sqrt{3}}{12} s ^{2}$ divide by area of square s^2 to get ratio $=\frac{12+7\sqrt{3}}{12}$

8. watchmath

dumbcow right, I forgot to multiplu by two here :tan 60 = 2s/(t-s) (I did s/(t-s) )