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Question 104946: HI
please help me with this
IS it possible to draw a triangle so that all three sides have positive slopes.If so do it ,if not explain why not.
Answer by kev82(151)   (Show Source):
You can put this solution on YOUR website! We can define a triangle by 3 points in the Cartesian plane. If (without loss of generality) we set one of them at the origin, that leaves us two point A and B, forming the triangle OAB. If we make sure we put the points A and B strictly in the first quadrant (x>0,y>0) then we know that side OA has positive slope, and so does side OB. Let us now assume that side OA has the smallest angle to the x axis. We need the line AB to have positive slope. By definition slope is the ratio of difference in y and difference in x, which will be positive if both the differences are positive. So as long as we choose B.x > A.x and B.y > A.y line AB also has positive slope. You can move the origin at will without changing slope.
Summarising.
1) Pick a point a (any point you like)
2) Pick a point b with x>0 and y>0
3) Pick a point c with x>0 and y>0
4) Let P=a, Q=a+b, R=a+b+c
5) Triangle PQR has to have all sides with positive slopes
This is sufficient but may not be necessary (something for you to think about). I.E. you may be able to find other triangles which have all sides with posotive slopes, but can't be written with P=a,Q=a+b,...
Kev
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