Darts and Kites

Explore the geometry of these dart and kite shapes!

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:

Let
$PS = x$
$SX= a$
$XR= b$
Can you relate the areas of the triangles using these values?