Grades 9-10 Video Solutions 2013-Level 9&10 Video Solutions 2013 problem15

Video Transcription

Question number 15, the diagram shows the rectangle A, B, C, D lying below the X-axis and to the left of the Y-axis with edges parallel to the coordinate axes.  For each point, A, B, C, and D, the Y-coordinate is divided by the X-coordinate.  Which of the points yields the smallest value  from this calculation?  Let's begin by picking a point and assigning it a coordinate  let's say the point B has the coordinate X sub B, Y sub B.  And then we observe that the ratio YB divided by XB would be exactly the slope of the line passing through the origin and the point B. And that line has the equation  in slope intercept form,  Y is equal to the ratio we are interested in as the slope times X. And then we can even write plus 0 for the X-intercept.  So the ratio we have to compare across the points is the slope  of the line connecting the point with the origin. As the points lie in quadrant three,  the coordinates are all negative.  The ratios will always be positive. And it's pretty clear that no matter of the size  and orientation and position of the rectangle, the point that is the left upper corner or here denoted as D would be the point that produces a line passing through it and the origin  that will always have the smallest slope.  So that is exactly the quantity we are looking for  and D is the answer to problem 15.

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