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Question:


dogcrazy
Student
High School - 10th Grade

What can you observe about the measure of the angle between two lines as you try make their slopes as close as possible?

For this question, I'm refering to a resource known as the Geometer's Sketchpad where you're able to construct and move around lines and angles. Even if you're not familiar with this particular program, just try to picture two perpendicular lines that intersect at point E. Then, try to imagine moving these lines so that their slopes are parallel.

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Posted by dogcrazy on Tuesday November 3, 2009 at 8:03 PM and tagged with geometry, math.

Answers:

  1. neela
    neela Teacher
    Graduate School

    eNotes Editor

    Let two lines be y=m1*x+c1 and y=m2*x+c2 in the slope intercept forms, where m1 and m2 are their slopes and c1 and c2 are their y intercepts .

    Therefore If A and B are the angles these lines make with X axis,and C is the angle of their intetsection,

    Then angle C =angle B-angle A.

    Therefore, tanC =tan(B-A) = ( tanA-tanB)/(1+tnA*tanB).......................(1)

    But tanA=m1, the slope of the line y=m1*x+c1 and

    tanB=m2, the slope of  the line y=m2*x+c2.

    Therefore, the equation (1) could now be rewritten like:

    tanC=(m2-m1)/(1+m2*m1)......................................(2)

    Therefore, when the two lines are perpendicular, C=90 degrees or

    tan C=(+or-)infinite. So, in the right side of (2) , the denominator should be zero , implying 1+m1*m2=0 or m1*m2=-1.

    As you move the lines to make the angle C less and less and then C=0 degrees, the the value of tanC=0. Or in the equation (2), the right side numerator is zero or m2-m1=0 or m1=m2.

    Hope this helps.

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    Posted by neela on Tuesday November 3, 2009 at 9:29 PM