16. GEOMETRIC ARITHMETIC INVESTIGATIONS

A. Proportionals

i. Given three segments a,b,c how do you construct the fourth proportional? (d so that a/b = c/d).

ii. If b and c are fixed, what is the locus of some variable point in your construction?

iii. Make a sketch if b and c are equal: in other words a:b = b: d.

iv. If a is fixed, how does d vary with b?

v. Make a sketch on sketchpad to show the locus of segments d such that a:b = b: d where a is fixed and b is a variable length.

B. Means and Extremes

Here is a geometric proof of “cross multiplication”: a/b = c/d if and only if ad=bc (for b, d = 0.)

Geometrically this theorem may be stated as relating size and shape: two rectangles ab and cd have the same shape, if and only if rectangles ad and bc have the same size.

i. ---> If a/b = c/d then ad = bc.

Suppose four segments a,b,c,d are in proportion a/b = c/d. Then put the two rectangles ab and cd vertex to vertex, and complete a larger rectangle, as shown.

Show that the areas of the two shaded rectangels bc and ad are equal.

ii. (converse) <--- If ad=bc, then a/b = c/d

Start with two rectangles ad and bc with equal area. Put them vertex to vertex and complete the picture, as below: the shaded areas are equal. Prove rectangles ab and cd are similar. (Hint: you must show that the diagonal line is a straight line!)

C. LAW OF COSINES