Problem Solving. Use Everything

We often need to use multiple problem solving techniques in the same problem. The important thing is to step back from the problem and try to work out which topics are involved and which techniques usually help when dealing with those topics.

Sometimes you will have to make mistakes and learn from them, giving you valuable experience.

Three Circles Example

Three circles each have radius $\frac{\sqrt{3}}{2}$ cm and centres A, B and C. The three circles are drawn so that they all touch each other once. Find the area of triangle ABC.

Solution

Clearly it would be a good idea to start with a diagram:

As we know the radius of each circle, we can add to the diagram the triangle we want to find the area of and then draw the triangle on its own for clarity:

There are several ways to find the area of this triangle, for instance to find its height, using Pythagoras’ Theorem on a right angled triangle with base $\frac{\sqrt{3}}{2}$ and hypotenuse $\sqrt{3}$ . Once we know the height, we can use the 1/2 x base x height formula for the area of a triangle.