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## High school geometry

### Course: High school geometry>Unit 4

Lesson 6: Proving relationships using similarity

# Proof: perpendicular lines have opposite reciprocal slopes

To prove that perpendicular lines have opposite reciprocal slopes, draw the two lines and label key points. By identifying the two right triangles that are formed, one can use the definition of slope to calculate the slopes of the two lines. Finally, using the properties of similar right triangles, show that the two slopes are indeed opposite reciprocals of each other.

## Want to join the conversation?

• Line DE is negative because it goes from D to E in the direction. He could have just gone in the direction of E to D and had line ED that doesn't need the negative symbol because its going in a positive direction! For that matter why did he make the line CB and not BC which would have needed the negative symbol? If any of the lines are changed as I mentioned then the proof would fail. So what am I not understanding? • If a line goes forward (to the right) in the upper right quadrant it has a positive slope, and if it goes to the lower right side it has a negative slope. This is reciprocated in the other direction (positive slope = bottom left and negative slope = top left). The ED/DE line goes to the right in the lower right quadrant, so the slope must be negative, unlike the BC/CB one which goes into the upper right quandrant.
• Why do perpendicular lines have a negative reciprocal slope ? I'm still a bit puzzled • At , Sal said that we have to take DE as negative then why didn't he took BC as negative on LHS? Isn't it also dropping? • He is saying DE is negative while AD is positive so that the slope of AE is negative, so he is using this fact for a purpose, not just to say that it is negative for no reason. On the other hand, if you start from A, you are correct that BC is negative, but AB would also be negative, so when you talk about the slope of AC, when you divide a negative by a negative, you would get a positive.
Hope this helps
• Isn't the similarity postulate supposed to be 'AA', instead of 'AAA'? • I think I finally understood how it worked when you drew the triangles. I get it now! When you look at one perpendicular line, and then you look at the second perpendicular line, it's almost as if the rise and run SWITCHED PLACES! That's why the slope of a perpendicular line is the negative reciprocal of the other perpendicular line!
Edit: That IS how it works, right? • How do you know how properly approach to the problem? I mean if I was going to prove such things I would go in the wrong direction and stuck there. Is there some common sense of proving things like this? Thanks! • Sometimes you can look for clues, patterns. Examine how you approached similar problems in the past. Other times, you got to pick a path and go through with it. The first person who came up with this particular problem didn't know what direction to go in, or even if there was any direction at all. If you get stuck, try a new angle.

If you try and try and try and try to do something, like how people tried to find a fractional representation of the square root of two, and it's just not working, maybe instead of trying to find a possible answer, you could prove why it's impossible. That's what Euclid did. He did something called a proof by contradiction, where you assume the contrary thing is true at the start of the proof. It's worth reading about. Hope this is enough of an answer to get you started, because really, this is what the joy of mathematics is all about.
• I could take the slope of L as the line AC and the slope will be negative not positive and the the slope Of M will not be negative reciprocal of L • Can we use dot products for deriving the same result?   