Question

the figure to the right represents a miniature golf course hole. the ball is at p and the hole is at q.
a. explain how to aim the ball so that it gets in the hole, if it is to bounce off one wall only.
b. explain how to aim the ball so that it gets in the hole, if it is to bounce off two walls.
m. aim the ball directly towards q
c. drop a perpendicular from q to \\(\overline{ab}\\). aim the ball at the intersection of the perpendicular and \\(\overline{ab}\\)
d. reflect p in \\(\overline{ab}\\) to get p. aim the ball at the intersection of \\(\overline{pq}\\) with \\(\overline{ab}\\)
b. explain how to aim the ball so that it gets in the hole, if it is to bounce off two walls. choose the correct answer below.
a. reflect p in \\(\overline{cb}\\) and q in \\(\overline{ab}\\). connect the images p and q by a straight line. aim the ball at the intersection of \\(\overline{pq}\\) with \\(\overline{ab}\\)
b. reflect p in \\(\overline{ab}\\) and q in \\(\overline{fa}\\). connect the images p and q by a straight line. aim the ball at the intersection of \\(\overline{pq}\\) with \\(\overline{ab}\\)
c. reflect p in \\(\overline{ab}\\) and q in \\(\overline{fa}\\). connect the images p and q by a straight line. aim the ball at the intersection of \\(\overline{pq}\\) with \\(\overline{fa}\\)
d. aim the ball directly towards q. it will then bounce off \\(\overline{cd}\\) and \\(\overline{ab}\\) on its way to q

Explanation:

For part a, when bouncing off one wall (say \(AB\)), the principle of reflection is used. Reflecting \(P\) over \(AB\) (denoted as \(P'\)) and then aiming at the intersection of \(\overline{P'Q}\) with \(AB\) works because of the law of reflection (angle of incidence equals angle of reflection).

For part b, when bouncing off two walls (\(AB\) and \(FA\)), reflecting \(P\) over \(AB\) (to get \(P'\)) and \(Q\) over \(FA\) (to get \(Q'\)), then connecting \(P'\) and \(Q'\) with a straight - line. The intersection of \(\overline{P'Q'}\) with \(AB\) gives the aiming point. This is based on the multiple - reflection property in geometry, where each reflection accounts for a bounce off a wall.

Answer:

a. D. Reflect \(P\) in \(\overline{AB}\) to get \(P'\). Aim the ball at the intersection of \(\overline{P'Q}\) with \(\overline{AB}\)
b. B. Reflect \(P\) in \(\overline{AB}\) and \(Q\) in \(\overline{FA}\). Connect the images \(P'\) and \(Q'\) by a straight line. Aim the ball at the intersection of \(\overline{P'Q'}\) with \(\overline{AB}\)