A fascinating math puzzle for you...?

Question

PQRS is a square of side 12cm which fits inside an equilateral triangle ABC in such a way that each of P,Q,R and S is on a side of ABC, as shown in this picture:

http://img248.imageshack.us/img248/2674/pic35bu.gif

Calculate the area of triangle ABC.

10 points goes to the correct answer with the clearest explanation. Thank you!

Answer 1

The area of an equilateral triangle is known to equal,

Area = (S^2 √3)/4

, where S is the length of a side. So all we need to do is calculate the length of one side to find the area of the triangle.

Angle A is 60° because the triangle is equilateral. Angle APQ is 90°, so we can calculate the length AP using

AP = PQ/Tan( 60°) = 12/√3 = 4√3

Calculating the length of SC is trivial; it equals the length AP. So now we can find the length of AC.

AC = AP + SC + 12 = 4√3 + 4√3 + 12 = 8√3 + 12

Plugging this value into the equation for the area of an equilateral triangle we get.

Area = ((8√3 + 12)^2 * √3)/4

Area ≈ 289.49 cm^2

Answer 2

I guess with the hieight of the triangle you are all set and it's clearly 12 + 12(\sqrt{3}/2}.

Answer 3

we have QR and AC are two parallel lines, so angle BQR = angle BAC = 60(degree), and we have angle QBR = 60(degree) therefore, triangle QRB is also a equilateral triangle.
That gives us QR = RB = BQ = 12cm
In triangle QPA, we have : sin(QAP) = sin(60 degree) = QP/AQ
=>AQ=QP/sin(60 degree)=12/[(square root of 3)/2]
Then AB = AQ + QB = 12cm + 24/(square root of 3) = ...(I dont have calculator here)
Finally, Area = 1/2*AB^2*[(square root of 3)/2] = ...
Hope it will help you!!!

Answer 4

AC is 24 cm long,hight is 12*root3
so the area is (1/2)(24*12*root3)