Assignment 5

Quiz #1 on Wednesday, Oct 14!

This will take only part of the hour, probably a proof and a construction.

Bring your transparent mirrors to class on Wed and after.

Assignment 5 (80 points, Due Mon. October 19)
Reading

B&B Chapter 5, also pp. 258-9.

GTC, Chapter 5

Main ideas

Concurrence of medians and connection with other midpoint figures.

Concurrence of angle bisectors and relation with inscribed circles.

Inscribed angles and other circle facts about chords, arcs, tangents from B&B.

Inscribed circles and angle bisectors and distance to lines

Regular pentagon properties and construction.

Introduction to Line Symmetry and Line Reflection

Problems to Turn In

In these proofs you can use theorems in B&B, Chaps 1-4 and the results in Chapter 5 which precede the problem.

5.1 Dimes (15 points)

(a) Do BB, p. 137, #8

(b) Given a circle of radius R, for what radius r will 7 circles of radius r just fit around the rim of the first circle? Find the value of r and then construct a Sketchpad sketch to check your answer. Print the sketch. Note: In (b), you can find an exact answer which is not in decimal form but in terms of trig functions; then you can evaluate this to get a numerical (approximate) answer.

5.2 Circumscribed quadrilaterals (15 points)

(a) BB, p 141., #8

(b) BB, p. 142, #9, #11

5.3. Angles in circles (10 points)

(a) BB, p 147-8 #5, #6

5.4 Tangent angles (20 points)

(a) BB, p 148 #7, #8, #9

(b) Explain how one can think of these facts as limiting cases of the previous facts in problem 3, when certain points coincide. (Tell which "tangent theorem" results when points coincide in "chord and secant theorems".)

5.5 Regular Pentagon (20 points)

(a) Carry out a construction of the regular pentagon as explained in B&B, p. 191.

(b) If the radius of the circle is 1, find and prove what is the length of the side of the inscribed regular pentagon. Then use this to justify the construction in (a).

Constructions (to do but not turn in)

Be prepared to construct

(a) Inscribed and escribed circles (incircles and excircles) of a triangle.

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