Do you think that all perpendicular slopes are related in the way you observed in problem 1-69? Investigate this idea by drawing three different lines with slope triangles on graph paper. Use slopes 1, 1/3 and 3/4 . Use tracing paper to rotate each line with its slope triangle 90º , either counterclockwise () or clockwise (). Find the slope of each new line. How does the slope of each rotated line compare to the slope of its original line? Share any patterns you find with your teammates. Use patterns from the work you have done to describe the general relationship of the slopes of perpendicular lines. That is, if you have two perpendicular lines and know the slope of one, how can you find the slope of the other?

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.