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From classics like Die Hard and Terminator to the legendary Kung Pow: Enter the Fist, action movies have provided countless hours of bated breath, wide-eyed stares, and unrealistic maneuvers that later become the subject of MythBusters episodes. (Ever wondered if it's actually possible to outfox a laser beam alarm system? Find out!) What beats a bucket of buttery popcorn and all the hand-to-hand combat, ginormous explosions, and obligatory car chase scenes that only action movies can provide?
Well, geometry doesn't have car chase scenes per se, but head-on crashes are far more common than you might think. You never expected math to be this action-packed, did you?
In geometry, we call two lines that crash into each other perpendicular, which is kind of the opposite of parallel. Not only do perpendicular lines intersect, they really, really intersect. We're talking a full-on crash that makes not one, not two, not even three, but four right angles.
Perpendicular lines make four right angles? Proving it is elementary, my dear Watson.
Given: Lines m and n are perpendicular.
Prove: ∠1, ∠2, ∠3, and ∠4 are right angles.
|1. Lines m and n are perpendicular||Given|
|2. ∠1 is a right angle||Definition of perpendicular|
|3. m∠1 = 90||Definition of right angle|
|4. ∠1 ≅ ∠3||Vertical angles theorem|
|5. m∠3 = 90||Definition of congruent angles, substitution|
|6. ∠1 and ∠4 are supplementary||Given in figure, straight angle theorem|
|7. m∠1 + m∠4 = 180||Definition of supplementary angles|
|8. m∠4 = 90||Substitution of (3) into (7)|
|9. ∠4 ≅ ∠2||Vertical angles theorem|
|10. m∠2 = 90||Definition of congruent angles, substitution|
|11. ∠1, ∠2, ∠3, and ∠4 are right angles||Definition of right angle using (3), (5), (8), and (10)|
So when two perpendicular lines intersect, we got something that looks like this.
Marking every angle with a 90° or right angle box is a little redundant though, so we usually just mark one of them. That's enough to know the rest of them are right angles, too.
Parallel lines get their own symbol, so it only makes sense that perpendicular lines get their own symbol too. We denote that lines FR and ED are perpendicular by writing FR ⊥ ED.
Perpendicular segments also have the peculiar quality of being the shortest distances from any point to another line. So if you're standing 20 feet away from a wall, that measurement is along a segment that's perpendicular to the wall.
It's also important to know that if two lines are both perpendicular to a third line, then those two lines are parallel to each other. So if lines f and g both intersect line b at 90° angles, then f and g are parallel. You can use your knowledge of corresponding angles, alternate interior angles, or alternate exterior angles to prove this.
Also, make sure you know that orthogonal means the same thing as perpendicular. Some teachers or textbooks like to use it because it makes them sound really fancy, like using the word "loquacious" instead of "talkative." They can use all the five-dollar words they want. We'll stick to ones that only cost a buck fifty and get a side of fries with the extra change.
Time to surf the x and y vibes and find out what the coordinate plane has in store for perpendicular lines. Sure, the x- and y-axes are perpendicular and those graph paper squares are totally perpendicular, but how do we find the equations of lines that are perpendicular to other lines? And we aren't talking your typical y = 3. We mean weird lines, with slopes of ¼ and y-intercepts like -23.
The secret to both parallel and perpendicular lines on the coordinate plane is the slope. Parallel lines have the same slope and they never intersect. Since perpendicular is sort of the opposite of parallel, we'd expect the slopes to be opposite some way. What we might not expect is that the slope is opposite in two ways.
For two lines to be perpendicular, one line has to have a slope that's the negative reciprocal of the slope of the other.
Okay, think of it this way. We've got a line y = 2x + 1.
We want a line that intersects this line at a right angle. How do we get that? For one thing, the line has to slope in the opposite direction, so let's try going from a 2 to -2.
They definitely intersect, but it still doesn't look quite right, does it? The angles formed aren't 90° angles…yet. Basically, to form a right angle, we need to get nitty gritty. A line with a slope of 2 goes up 2 units on the y-axis for every 1 on the x-axis.
A perpendicular line should go down 1 unit for every 2 units on the x-axis. This means a slope of not just -2, but -½. That's the negative reciprocal.
So let's take a really weird line like y = ¼x – 23. A perpendicular line would be y = -4x – 23. Or y = -4x + 1, or y = -4x + 1001. The y-intercept doesn't really matter because it only changes where the two lines intersect, not how.
Find a line perpendicular to the line 2x + y = 7 that goes through the point (4, 5).
We've gotta find the slope of the line, so let's change this sucker into slope-intercept form. It'll look something like y = -2x + 7. Actually, it'll look exactly like that.
After we pluck the slope -2 out from the equation, we need its negative reciprocal. The reciprocal means flipping the number so its numerator and denominator are switched, and then taking its negative. In this case, we end up with positive ½. Two wrongs might not make a right, but in this case, two negatives certainly help that process along.
To get the new equation, we start with y = ½x + b. To find what b equals, we'll need to plug in that point (4, 5).
y = ½x + b
5 = ½(4) + b
5 = 2 + b
b = 3
So, our perpendicular line is y = ½x + 3 or 2y – x = 6.
Are x + y = 5 and x – y = 5 perpendicular?
In other words, this question asks whether or not the slopes of these two lines are negative reciprocals of each other. That's really all it takes for two lines to be perpendicular, right?
In slope-intercept form, x + y = 5 turns into y = -x + 5, which has a slope of -1. The other equation, x – y = 5, turns into y = x – 5, which has a slope of 1.
So, is 1 the negative reciprocal of -1? You betcha. They're as perpendicular as can be.