A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?
Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).
How can you make an angle of 60 degrees by folding a sheet of paper
Correct solutions from Mary of Birchwood High School and Andrei
of School 205 Bucharest have contributed to the edited solution
As triangle DEF is equilateral, all its angles are $60$ .
Angle AEF $= 180 - 60 - c = 120 - c$
Angle BFD $= 120 - b$
Angle EDC $= 120 - a$
From triangle FAE, I calculate angle A:
As triangle ABC is an isosceles Angle ABC $=$ Angle ACB $=
Therefore $180 - (120 - b + a ) = 180 - (120 - a + c )$
Therefore $60 + b - a = 60 + a - c$
Therefore $2a = b + c$
Therefore $a = (b + c)/2$
Angle ABC $= 180 - (120 - b + a ) = 180 - (120 - a + a ) =
Angle ACB $= 180 - (120 - a + c ) = 180 - (120 - a + a ) =
Therefore angle BAC is $60$
Therefore triangle ABC is equilateral.