METU Department of Mathematics Math 303 History of Math Concepts Fall 2012

Midterm 2

December 10 2012, 17:40 – 19:00

Name and the student number:

Part II (9 questions, 29 points, 40 minutes)

Q1(4pts). Describe the fifth Postulate from ”Elements”. Give at least two more claims equivalent to this Postulate.

Q2(4pts). Where is the center of mass of a triangle ABC, if the weights located at A,B, and C are 1,2, and 6 respectively ?

Q3(3pts). Formulate the results of Archimedes concerning quadrature of a sector of parabola and a triangle inscribed into a parabola (“Archimedean triangle”).

Q4(3pts). What is a semiregular polyhedron ? Who found the number of such polyhedra and what is this number ?

Q5(2pts). What is the definition of a circle by Apollonius ?

Q6(4pts). What can be the number of normals to an ellipse, and how does it depend on the position of a point ? Sketch the evolute of an ellipse and explain its meaning.

Q7(3pts). Formulate the Appolonius Problem. Give a sketch.

Q8(3pts). Apply Heron’s formula to find the area of a triangle with the sides of length 7,8, and 9.

Q9(3pts). Who studied the problem of maximal area of polygons with a fixed perimeter ? What was the result obtained ?