Describe the locus defined by
A pdf version.
Monthly Archives: March 2024
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.
Midpoint
I find many students struggle with the formal notation for calculating midpoint.
The idea of finding halfway between the x and y co-ordinates seems to resonate.
You can then introduce the formula and use that as a way of getting to grips with the distance formula and the gradient formula.
My notes on midpoint.
My notes on gradient.
More on gradient distance between points later.
Filed under Co-ordinate Geometry