Darts and Kites

A rhombus PQRS has an angle of 72 degrees. OQ = OR = OS = 1 unit. Find all the angles, show that POR is a straight line and that the side of the rhombus is equal to the Golden Ratio.

Pent

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.

Pentakite

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.

Golden Thoughts

Stage: 4 Challenge Level:

Sue from Madras College, St Andrew's, Scotland sent in a good solution to this problem.

Let $SX = a$ and $XR= b$ then $PQ=a+b$ and, since the areas of the triangles $SPX, XYR$ and $PQY$ are equal, we know that $YR= ax/b$ and $QY = ax/(a+b)$. Because $PS = QR$ we now have $$\frac{ax}{a+b} + \frac{ax}{b} = x$$ and simplifying this we get $b^2 -ab - a^2 = 0$. So, writing $t$ for $\frac{b}{a}$, we get the quadratic equation $$t^2 - t - 1 = 0$$ which has roots $$t = \frac{1 \pm \sqrt 5}{2}$$ but we know that $t$ is positive so $$t =\frac {1 + \sqrt 5}{2}.$$ Also, writing the ratio $RY/YQ$ in terms of $t$ we get $$\frac{RY}{YQ} = \frac{a+b}{b} = \frac {1}{t} + 1.$$ We already know that $t=\frac{1}{t} + 1$ because $t^2 = 1 + t$. So $$\frac{RX}{XS} = \frac {RY}{YQ} = \frac {1 + \sqrt 5}{2}.$$