You may also like

problem icon

Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..

problem icon

At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

problem icon

Six Discs

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

Cosines Rule

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

(a) Prove that a triangle with sides $3$, $7$, $8$ contains a $60$ degree angle.

(b) Three points $A$, $B$ and $C$ lie in this order on a line, and $P$ is any point in the plane. Use the Cosine Rule to prove that:

\[{AP^2\over AB.AC}+{CP^2\over CA.CB} = 1 + {PB^2\over BA.BC}.\]

Triangle.