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Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface of the water make around the cube?

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Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

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Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

Rati-o

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Correct solutions were received from Chen from The Chinese High School Singapore, Andrei of School 205 Bucharest, Mary of Brirchwood High School (whose solutions is used as the basis of the one below) and Sana, Jenny, Chris and Rosion of Madras College, St. Andrews.

Diagram showing l and w as one third of the length and widthl is a 1/3 the length of the rectangle

w is a1/3 the width of the rectangle

area of the rectangle = 3l x 3w = 9lw

Area triangle A = Area of triangle C = lw

Area triangle B = Area of triangle D= lw

Therefore area of the parallelogram = 9lw - 4lw = 5lw

Therefore ratio of the area of the parallelogram to the area of the rectangle is 5lw : 9lw = 5:9