Assignment 3 (50 Points) DUE MONDAY 10/9
Study carefully chapters 2 and 3 in B&B. For logic and proof, you may wish to return to chapter 1 from time to time.
3.1 Point Symmetry on a Line (10 points)
Definition: Given two points A and P, the point reflection of P in A is the point P' such that A is the midpoint of P and P' (in the special case that A = P, we set P' = P). Thus point reflection with center A can be thought of as a function taking point P to point P'.
Problem: Given 3 points A, B and P. Let P' be the point reflection of P in A and let P'' be the point reflection of P' in B. Now suppose that A, B and P are all on a line m with a given ruler. (a) If the ruler numbers of A, B and P are a, b, x, find the ruler numbers y of P' and z of P''. In other words, find a formula for P' and P'' in terms of a, b, x. (b) Compute the distance from P to P''. How does this distance change when P changes? How does this distance depend on the distance from A to B?
Note: The original handout of this problem had an error in the definition of point reflection, leading to possible confusion. To make sure everyone has the correct problem and solution, the solution is linked here.
3.2 Some Protractor Formulas (10 points)
(a) Given a point O and a protractor on the rays with endpoint O (measuring in degrees). Let ray OP be a ray with endpoint O that has protractor angle p. If d is a number, 0 < d < 180, find the protractor angles a and b of two rays OA and OB that make an angle of d with OP (i.e., Ð POA = d and Ð POB = d). Explain why these are the only rays that make angle d with OP.
(b) Continuing with the rays with endpoint O, let OC and OD be rays with protractor angles c and d. Find the protractor angle of the angle bisector.
Comment: This is a bit tricky; it is a somewhat more complicated than a single formula. Do some examples for make sure you understand the whole picture. Check that for a protractor running from 0 to 360, your answer works in the cases: (1) c = 20 and d = 80 and (2) c = 20 and d = 330.
3.3 Construction: Copying an angle. (5 points)
Read the hints for copying an angle with straightedge and compass on page 173 of B&B. Write a set of the directions for this construction and then explain why it works (informal proof).
3.4 Construction: Rational Numbers (5 points)
Draw a segment AB of length 6 inches. Without measuring, use a straightedge and compass to construct a point C so that AC/AB = 3/5.
3.5 Similarity of right triangles (10 points)
B&B page 85, #12 and #13
3.6 Subtriangles of a right triangle (10 points)
Let triangle ABC be a right triangle, with angle C being the right angle. Let D be the foot of the altitude through C (i.e., D is the point on AB so that line CD is perpendicular to line AB). If the lengths of the sides of triangle ABC are |AB| = c. |BC| = a, |CA| = b, find the lengths of segments CD, AD, and BD (in terms of a, b, c). Give a proof for your answer.