How Do You Prove That Perpendicular Lines Have Negative Reciprocal Slopes?

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SUMMARY

The discussion centers on proving that if line L1 is perpendicular to line L2, then the slope of L2 is the negative reciprocal of the slope of L1. Participants clarify that perpendicular lines intersect at 90-degree angles and that the slope of L1, denoted as M1, cannot be zero. The proof requires demonstrating that the product of the slopes of two perpendicular lines equals -1, leading to the conclusion that the slope of L2 is -1/M1.

PREREQUISITES
  • Understanding of basic geometry concepts, specifically perpendicular lines.
  • Knowledge of slope calculation in coordinate geometry.
  • Familiarity with the concept of negative reciprocals.
  • Ability to work with linear equations in a Cartesian plane.
NEXT STEPS
  • Study the properties of perpendicular lines in coordinate geometry.
  • Learn how to calculate slopes of lines given two points.
  • Explore the relationship between slopes and angles in geometry.
  • Investigate proofs involving inner products and vector definitions of perpendicularity.
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Students studying geometry, mathematics educators, and anyone looking to understand the relationship between slopes of perpendicular lines in coordinate systems.

STAR3URY
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Please Help With A Proof!

If L1 is perpendicular to L2, prove that the slope of L2 is the negative reciprocal of the slope of L1..


PLEASE HELP ME!
 
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What is the definition of perpendicular? definition of slope? Any ideas?
 
EnumaElish said:
What is the definition of perpendicular? definition of slope? Any ideas?

Perpendicular is when two lines intersect to form 90 degree angles, slope is whether the line is moving up or down, and how much like the top value is the y and the bottom is the x.
 
If I tell you that the tangent of the angle that a line makes with any horizontal to it gives you the gradient. does that help?
 
Draw the right triangle connecting the points (0,0), (x,0) and (x,y). Got it? So the hypotenuse has slope y/x. Now rotate it 90 degrees and tell me what the endpoint of that hypotenuse is? What's the slope of that hypotenuse?
 
My professor doesn't want us to use trigonometry because we didn't learn that yet..and Dick, I am not sure I am allowed to use that as a proof..
 
STAR3URY said:
My professor doesn't want us to use trigonometry because we didn't learn that yet..and Dick, I am not sure I am allowed to use that as a proof..

If you can't rotate a point 90 degrees, then what's your definition of perpendicular?
 
Dick said:
If you can't rotate a point 90 degrees, then what's your definition of perpendicular?

During class all he told us was to prove that if
1). L1 is perpendicular to L2
2). slope of L1 is M1 where M1 can't be 0

prove that slope of L2 is -1/M1...thats it and he said not to use anything we didn't learn, he didnt talk about rotating anything, or trignometry. =(
 
If you don't know that the line through the point (x,y) and (0,0) and the line through the point (-y,x) and (0,0) are perpendicular then I don't think we are going anywhere. What IS your definition of perpendicular?
 
  • #10
Well i can't use my definition all he said was his definition, and that's that they intersect and the slope of l2 is negative recipricol and that's it...we have to prove that
 
  • #11
STAR3URY said:
Well i can't use my definition all he said was his definition, and that's that they intersect and the slope of l2 is negative recipricol and that's it...we have to prove that

Oh, come on. You said the definition of perpendicular is that the lines meet at 90 degree angles. This means you can use some geometry. Use it.
 
  • #12
Did you learn to define perpendicular in terms of inner products (of two vectors)?
 
  • #13
STAR3URY said:
Well i can't use my definition all he said was his definition, and that's that they intersect and the slope of l2 is negative recipricol and that's it...we have to prove that
That makes no sense. Apparently you want to PROVE that his definition is... what? Equivalent to some other definition, apparently! You can't PROVE that his definition is correct using only his definition! What is YOUR definition of perpendicular? That two lines intersect at 90 degrees?
 
  • #14
In graph (orthonormal system) we know that if two lines are perpendicular then slope 1*slope 2 = -1 (it is a rule)
so slope 1 = -1/slope 2
 
  • #15
mohammad_selek said:
In graph (orthonormal system) we know that if two lines are perpendicular then slope 1*slope 2 = -1 (it is a rule)
so slope 1 = -1/slope 2
One man's definition can be another's problem. :smile:
 
  • #16
What is the lesson you took ? to determine the proof ?
 

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