One of my year 10 students came with this question from his text book.
Here is my solution.
A pdf version of the solution
Category Archives: Trigonometry
Right-Angled Trigonometry Question (very hard)
Filed under Right Trigonometry, Trigonometry, Uncategorized
Three Circles – Area Problem
This is question 5 from the UK Maths Trust Senior Challenge October 2023.
I have tackled this in three ways; using non-right trig to find the area, Heron’s Law, and the Shoelace Formula.
Method 1
Use the area of a triangle formula
Use the cosine rule to find cosθ.
Once we have cosθ, we can find sinθ.
Hence the area is,
Method 2
Use Heron’s law.
Heron’s law is a way of calculating area of a triangle from the lengths of the three sides of the triangle.
This is my preferred method – simple and direct.
Method 3
Shoelace formula (Gauss’s Area formula)
We need to allocate each of the vertices a co-ordinate.
The co-ordinates are listed in an anti-clockwise direction.
This is probably a bit over the top, but once you get the hang of it, it’s very easy.