20 Feb 2013

Midweek Maths: Two Diamonds

We continue on the theme of geometry. Like the previous question, the one below can be done in different ways: with pure trigonometry; with geometric transformations; and even using coordinate geometry.

This is true of many questions in mathematics. An important skill in mathematics is not just the ability to solve problems, but the capacity to quickly go through different possible methods and to pick the most efficient. This is a skill that is difficult to teach but essential in being able to save time in a mathematics competition – time that will be precious for the more challenging questions. One way for you to measure the most efficient method is to actually do the question below in three different ways! Try it and see for yourself which method seems faster.

It is also important to develop some self-knowledge about which branches of mathematics you find come most naturally. Some students prefer number theory whereas others may like geometry or probabilities. In team competitions it is vital to distribute each given question to the person who is the ‘expert’ in that field. Sometimes, the best team is not always the sum of the best individuals.

The Question

Below are two identical unit squares. Within them are drawn lines starting from a corner to a mid-point of a side. Two different layouts are shown and two quadrilaterals, A and B, are highlighted in grey. Find the ratio of (Area A) : (Area B).

(This question has been adapted from a UKMT SMC.)

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