Reading: Read Chapter 5 of BB, noting the statements of all the theorems carefully and also the statements of the starred problems. We have already proved most of these theorems – the ones about tangents and theorem 22 about inscribed angles – in lab. Also study the chapter summary.
We will define a triangle ABC to be a golden isosceles
triangle if there is
a point D on segment AB such that AB = AC and CB = CD = AD.
a) Prove
that AB/BC = p = (1 + sqrt 5)/2.
(This ratio is called the golden ratio and is sometimes denoted by the
Greek letter phi.)
b) Suppose
that RSTUV is a regular pentagon.
Draw all or some of its diagonals, RT, TV, etc. Find a golden isosceles
triangle in the figure and use it to prove that the ratio of a diagonal to a
side such as RT/RS = the golden ratio f.
c) Prove
that your candidate for a golden isosceles triangle satisfies the definition of
such a figure in the first sentence of this problem.
Note 1: We did a lot of this in class, but you
are being asked to write up that work carefully. So you can assume the usual theorems about similar
triangles, angle sum, etc. But you
can't assume what you showed about pentagons in class without proving it
here. A regular polygon is defined
in BB.
Note 2: You are not asked to construct these
figures in this problem. However, you are asked to construct a regular pentagon
in Construction Portfolio 3.
Prove BB p. 127 #4.
Read and figure out why all the starred problems on BB page 137 are true. Write down and turn in proofs for #5 and #6.
Let A, B, C, and D be points on a circle c so that quadrilateral ABCD is inscribed in c. Suppose the chords AC and BD intersect at a point P inside the circle.
a) Prove that, in the quadrilateral ABCD, opposite angles are supplementary.
b) The diagonals divide ABCD into four triangles ABP, etc. Prove that each triangle of these four is similar to one of the others.
c) Use the results of (b) to prove a relationship among PA, PB, PC, and PD.