Assignment 2A (Due Wednesday, 10/8)

2.1 Circle inscribed in an angle

Consider an angle ABC and a circle with center O that is tangent both to ray BA and to ray BC. Prove that O is on the angle bisector of ABC.

Hint: Create a good drawing and label important points. Decide what you need to prove about angles, lengths, etc. to demonstrate the result. Then use your tools to prove some angles, lengths, etc., are congruent.

2.2 Midpoints of midpoints

Suppose that A, B and C are points on a line m. Also suppose that the line has a ruler that makes each point correspond to a number. Denote by a, b, c the numbers that correspond to A, B, C.

• Let D be the midpoint of AB and E the midpoint of BC. If the numbers d and e correspond to D and E, find formulas for d and e in terms of a, b, c.
• Let F be the midpoint of DE. If the number f corresponds to F, find a formula for f in terms of a, b, c.

2.3 Point Reflection

Again on the line m with a ruler that corresponds points to numbers, suppose that point A corresponds to 2 and point B corresponds to -1.

• According to the definition in Brown, if P is a point, the point reflection of P with center A is the point P' so that A is the midpoint of PP'. If the point P corresponds to number x, then what number corresponds to P'?
• If P'' is the point reflection of P' with center B, what number corresponds to P''.
• Repeat the last two exercises of computing the numbers corresponding to P' and P'' when A correponds to 77 and B corresponds to 74. What is the same and what is different? Given a good explanation why this is so.