You may also like

problem icon

So Big

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

problem icon

30-60-90 Polypuzzle

Re-arrange the pieces of the puzzle to form a rectangle and then to form an equilateral triangle. Calculate the angles and lengths.

problem icon

Bend

What is the longest stick that can be carried horizontally along a narrow corridor and around a right-angled bend?

Gold Again

Stage: 5 Challenge Level: Challenge Level:1

The first part of this solution came from Dorothy, S4, Madras College, St Andrew's and the second part from Vassil, Y11, Lawnswood High School, Leeds.

Consider the rhombus as illustrated, where $x$ is an unknown length. We have to find the value of $\cos 36^\circ \cos 72^\circ .$

I filled in the remaining angles and lengths, showing triangles $PCB$ and $PCD$ to be isosceles triangles with angles of $108^\circ$, $36^\circ$ and $36^\circ$ and sides $PC = PB = PD = 1$ unit.

fig1 I found $\cos 36^\circ$ and $\cos 72 ^\circ$ by using the cosine rule for triangles DCP and APD respectively. \[ \cos 36^\circ = {{x^2 + 1 - 1}\over 2x} = {x\over 2} \] \[ \cos 72^\circ = {{1 +x^2 - x^2}\over 2x} = {1\over 2x} \] Combining these two expressions, \[\cos 36^\circ \cos 72^\circ = {x\over 2}.{1\over 2x}={1\over 4}.\]

Consider the area of the triangle `above' the diagonal, and express it is the sum of two areas : \[{1\over 2} x^2 \sin 108^\circ = {1\over 2} x\cdot1\cdot\sin 72^\circ + {1\over 2} 1\cdot 1\cdot\sin108^\circ.\] As $\sin 108^\circ = \sin 72^\circ$, this gives $x^2=x+1$ and hence $x$ is the golden ratio: \[x = {1+\sqrt{5}\over 2} = \varphi .\] Hence \[\cos 36^\circ - \cos 72^\circ = {x\over 2} - {1\over 2x} = {x^2-1\over 2x}={1\over 2}.\]

Let the common side of the two triangles be $x$. Then we have: $b = x\cdot \cos 72^\circ$ and $ a= {x\over{\cos 36^\circ}}.$ Therefore \[{b\over a} = \cos72^\circ \cos 36^\circ = {1\over 4}\] so $a$ is four times bigger than $b$.


Here we have ${{x + b}\over a} =\cos 36^\circ$ and ${x\over a} = \cos 72^\circ$ so \[{b\over a} = \cos 36^\circ -\cos 72^\circ = {1\over 2}.\] Therefore $a$ is twice the length of $b$. fig 3