Exercises 1

1. Devise ways of using straight edges/compasses (or stretched strings!) to carry out the following tasks:
   
   1. Bisect a given line segment;
      
   2. Bisect a given angle;
      
   3. Construct a line a right angles to a given line through a point on that line;
      
   4. Drop a perpendicular to a given line from any point not on the line;
      
   5. Draw a line through any point parallel to a given line;
      
   6. Constuct a circle through any three non-collinear points;
      [The circumcentre is the meet of the perpendicular bisectors of the sides]
      
   7. (*) Divide a given line segment into three equal pieces.
      
   8. Given a segment of a line, extend it at one end;
      [You probably needed that earlier without realising it!]

   Solution to question 1

2. Prove that the set of rigid motions of the plane R² forms a group.
   Two figures in the plane are called congruent if there is a rigid motion mapping one to the other. Prove that figures congruent to the same figure are congruent to each other.
   (If you know what the word means) deduce that congruence is an equivalence relation.

   Solution to question 2

   

3. In the diagram shown the three triangles A, B and C are congruent. Find a rotation about some point of the plane which maps A to B.
   Find one that maps B to C and one that maps A to C.

   Solution to question 3
   
   

4. The corners of a tile in the form of an equilateral triangle are labelled A, B, C anticlockwise. Describe the result of rotating the tile anticlockwise by π/3 (= 60°) about A, then by the same amount about B and C ?
   What happens if you rotate by this same angle first about A, then about C and finally about B ?

   Solution to question 4

5. If f is a bijection from the plane to itself which preserves all lengths
   
   (i.e. d(x, y) = d(f(x), f(y)) for all x, y)
   
   show that f also preserves all angles.
   Give an example of a bijection which preserves all angles but not lengths.

   Solution to question 5

   Prove that the determinant map is a group homomorphism from the group GL(n, R) of all invertible n × n real matrices to the multiplicative group R - {0}.
   Deduce that SL(n, R) is a normal subgroup of GL(n, R).

   Solution to question 6

   Let A be an n × n matrix. Prove that A is orthogonal (i.e. has orthonormal columns -- of length 1 which are mutually orthogonal) if and only if A^tA = I.
   Deduce that for such a matrix AA^t = I and that A also has orthonormal rows.
   Prove this last fact directly for the 2 × 2 matrix by showing that if
   a² + c² = b² + d² = 1 and ab + cd = 0 then ac + bd = 0 and a² + b² = c² + d² = 1.

   Solution to question 7