A fellow maths teacher (@sitaylor7) at my school introduced me to this little curiosity. Since investigating, I believe that full credit should go to Rene Descartes although I’m not entirely certain he was the originator.

Draw a line segment from A to B of length x. Add one to its length to make a straight line ABC. Now draw a semi-circle from A to C. The vertical height from B to the intersection point between the line and semi-circle will be √x. How neat! I was desperate to proof it and after doing so thought it would be a nice activity for my Year 12 and 13 Further Maths classes. I got them on Geogebra and explained the set up without mentioning the square root aspect, so that they were to determine the significance of the vertical height.

Their Geogebra skills are getting better and they were confident to give it a go. They chose a random value for x and started investigating. Some immediately knew to test and record the vertical height for different values of x, where as some were trying to determine the significance based on one result. One pair had chosen an x value of 10, and so their height was √10 or 3.16 and so thought that may π be involved!

One student went about plotting x vs the height and tracing its path as x varied. This trace mapped the movement of the intersection point of the semi-circle and vertical line and was of course the curve y = √x.

Once this relationship had been established, the students went about trying to prove it. They found it difficult but with a little guidance managed the proof using Pythagoras in a couple of different ways. Here is the working of one student:

The idea of using similar triangles to prove the result did not come to the students until prompted. I find that generally the teaching of similar triangles (myself included) can be rather routine and doesn’t give the students an appreciation of the usefulness that applying this technique can have. Next time I teach similar triangles I will endeavour to use more geometric problems in non-standard forms to develop students’ use of this resource.

The proof below (courtesy of ‘mau’ from the maths exchange website) demonstrates the method. The students were impressed and surprised by its simplicity and I overheard “Oh wow…that’s cool”.

One or two of my students went further and explored the situation in which the additional length added is not just one, but β. I was really impressed by this thinking and the proof the student showed me – after a quick re-draft below. He uses a Pythagorean approach, although again, the similar triangle approach would have arguably been more efficient.

So, an enjoyable couple of hours exploring a nice bit of geometry. If you have any further ideas, questions or if you try it with one of your classes then please get in touch!

October 21st 2018 Update: New and quicker method found!

Using the intersecting chords theorem, this relationship comes out very quickly and elegantly. Most of my sixth formers were impressed with this and wondered why they were not exposed to this when looking at circle theorems? See below: