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March 9, 2013

David asks…

How do you write a proof of the slope formula y2-y1/x2-x1?

Can someone write a step-by-step thing of how to prove the equation y2-y1/x2-x1? that helps create the slope of a line?

Administrator answers:

You define slope of a (non-vertical) line as follows: let (x1,y1) and (x2,y2) be any two points on the line, then the slope of the line is the number (y2-y1) / (x2-x1).

Although this is a definition there is still something that must be proved. Namely, suppose you pick two other points on the line and use them to calculate slope. How do you know that you will get the same answer? You must prove that you do. In other words you must prove that the definition is “well defined”.

To do this you draw two right triangles, both with legs parallel to the axes and with hypoteneuse the line segment of the line joining two points you used to calculate a slope. You must find a geometric proof that these two triangles are similar. Then, you invoke the theorem that ratios of corresponding parts of similar triangles are equal. That will show that the two calculated slopes are equal and hence that the definition is “well defined”.

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