Answer **two** questions, one of which **must be question 4**.

- (a) In Book 2 of Euclid's elements he dealt with geometrical algebra. His proposition 4 was "If a straight line can be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments
Use a diagram to illustrate that this is the same as

(a + b) = a + 2ab + b

(b)Describe Pythagoras' Theorem with Euclid's proof (use non technical language to describe how the proof works)

- Discuss Euclid's Books V VI XI XII and explain congruency similarity and conics (briefly!)
(Mathematics in Western Culture)

- Find 2 more methods of proving Pythagoras' theorem (there are approximately 257!!)
- If the Greek scholar were required to construct a line x having the property expressed by
ax - x = b

Where a and b are line segments with a>2b, he would draw line AB = a and bisect it at C. Then at C he would erect a perpendicular CP equal in length to b. With P as centre and radius a/2 he would draw a circle cutting AB at D.

Then on AB he would construct rectangle ABMK of whidth BM = BD and complete the square BDHM. This square is the area x having the property specified in the quadratic equation. Try this construction (choose values of a and b)

Why does this construction work?

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