Mathematics
Intermediate Mathematical Challenge
2024
24

24

The diagram shows a regular hexagon RSTUVW.

The area of the shaded pentagon RSTPQ is one quarter of the area of the hexagon. Jay and Kay walk around the hexagon from P to Q, Jay going clockwise and Kay anticlockwise.

What is the ratio of the distance Jay walks to the distance Kay walks?

Answer
SOLUTIOn

Let O be the centre of the hexagon, and let M and N be the midpoints of UT and W R, respectively.

We also let s be the side length of the hexagon.The three diagonals through O together with the lines UW, MN and TR divide the hexagon into twelve congruent triangles.

Let t be the area of one of these triangles. Then the area of the hexagon is 12t.

The area of the shaded pentagon RSTPQ is one quarter of this.

So its area is 3t.The pentagon RSTPQ is made up of the triangle RST and the rectangle QRTP. We see from thediagram that the area of RST is 2t. Hence the area of the rectangle QRTP is t.

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