## Cycling Distances

GCSE Text:
A cycle path has been built into a hill, where cyclists travel down a  ramp, turn a corner, then travel down a second ramp to reach the bottom of the hill. In this diagram, cyclists travel from A to B, then from B to C. Calculate the total distance that the cyclist travels on the path, to travel from the top of the hill to the bottom.

Suggestions:
Students often suggest ‘Pythag’ quite quickly after recognising that the hypotenuse is to be calculated, but then struggle to identify 2 separate triangles. It is useful to suggest covering up sections of the diagram to reveal familiar shapes, or annotating/modifying the diagram on a GCSE paper to add a horizontal line to create an upper distinct triangle. It is also useful to discuss estimation of a hypotenuse given the size of the two sides, in order to avoid silly errors (like forgetting to root the sum).

Extension Ideas:
Teachers could extend the learning by considering:

• Adding a new point ‘D’ (to the right of the bottom right corner) which offers a ‘steep’ quick way to get to the bottom of the hill. This path will be 40m in length (the hypotenuse), so ask students to calculate the horizontal distance of the new finish point D.
• If students know trigonometry, this is a natural lead-in to calculating the angles of the cycle paths in the diagram. Which section is steeper – A to B or B to C?