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Golden Triangle

Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.

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The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

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ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

Road Maker 2

Stage: 5 Short Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

This problem follows on from Road Maker, where the rules of making roads are detailed in full.

The Munchkin road making authority have commissioned you to work out the possible destinations for their roads. Use Cartesian coordinates where the first tile is placed with opposite corners on $(0,0)$ and $(1,1)$.

Investigate ways in which you can reach your destination. You may like to consider these questions:
  1. Can you make roads with rational values for the $x$ coordinate of the destination?
  2. Can you make roads with rational values for the $y$ coordinate of the destination?
  3. Can you create a road with the $x$ coordinate equal to any integer multiple of one half?
  4. Can you make roads for which the coordinates of the destination are both rational? Both irrational?
  5. Can multiple roads lead to the same destination? For which destinations is the road unique?

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