1.) A cylinder and a cone have the same height, but the base radius of the cylinder is only half that of the cone. If the volume of the cylinder is 1, then what is the volume of the cone?

2.) Quadrilateral ABCD satisfies both conditions below:

• The translation that maps A to D also maps B to C.

• The reflection over line BD maps A to C.

Bob claims that quadrilateral ABCD must be a square. Do you agree with him?

If yes, prove that quadrilateral ABCD must be a square.

If no, provide a counterexample to Bob’s claim and explain why the counterexample satisfies both conditions.

3.) In triangle ABC, sin A: sin B: sin C = 2 : 3 : 4. Find cos(A+C)

Please explain if you can! :)

Treetrunk May 27, 2021