Answers to Quiz 3

Problem 1. Compute a distance (10 Points)

In the plane, A' is on ray OA and B' is on ray OB. The lengths are |OA| = 7, |OA'| = 3/7, |OB| = 5, |OB'| = 3/5, |AB| = k, for some constant k.

• Find the distance |A'B'|. Show your reasoning briefly. (The answer should only involve k and numerical constants – no other symbols.)

Triangle AOB is similar to triangle B'OA' by SAS, since the shared angle AOB = angle B'OA'. Also, OB'/OA = 3/35 = OA'/OB. The ratio of similitude from AOB to B'OA is 3/35, so |A'B'|/BA| = 3/35. Thus |A'B'| = 3k/35.

Problem 2. Construct Geometric Mean (10 Points)

a) What is the definition of the geometric mean of two numbers a and b?

The geometric mean of a and b is the square root of ab.

Construct a segment CD whose length is the geometric mean of the two lengths a and b. List BRIEFLY your main steps so that one can follow what you did.

Problem 3.

Part A. Construct by reflecting across c first, then d. If the order is backwards one gets the wrong D1.

Part B. The most efficient construction is to notice that D2 = D1, so the point is already constructed. This is true because the angles between c and d and a and b are congruent.

(In principle one could do this less efficiently get D2 by reflecting across a first, then b. But since this first reflection image will not be on the paper, this is not an efficient construction, or even practical in this case.

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