# Grades 8 to 12 A Mathematical Lens

*By toni On 2 July, 2019 · Leave a Comment*

**STEP 1**

**1a) **Double click on the picture above to start the movie showing how to draw a circle using the paperclip compass.

**1b)** Draw 2 circles of the same radius with centres A and B so that each circle goes through the centre of the other circle.

**1c) **Explain how you know that triangle ΔAEB is equilateral.

**1d) **Suppose AE = 1 unit. Give the length of EF leaving the answer as a surd.

**1e) **Using a calculator work out the length of EF to one decimal place with your value of AE. Now measure EF in your diagram. Do your two answers agree?

**STEP 2**

**2a)** Draw the extra lines in your diagram as shown here and label all the points.

**2b)** Calculate the lengths CA and CD for circles with AE = 1 unit giving your answers as surds.

**2c)** Using a calculator work out the lengths of CA and CD to one decimal place with your value of AE. Now measure CA and CD in your diagram. Do your measurements agree with your answers?

**STEP** 3

**3a)** You have worked out the radius of the pink disc in STEP 2 and now you can draw the circles around the pink and grey discs in your own diagram.

**3b)** Calculate the length BG with AE = 1 unit leaving your answer as a surd.

**3c)** The Golden Ratio φ = ½(1 + √5) is a very important number in mathematics. Have you found that BG = φ – 1? If not go back and check your working in STEP
3b).

**STEP 4**

Find out from the internet what what you can about this mathematical lens, so called because it is shaped like the human eye. The diagram, called Vesica Piscis, appears in many ancient mathematical writings and it has been considered a sacred symbol for thousands of years.

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