WEBVTT
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One of the places where algebra and geometry
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really overlap is in the Cartesian coordinate plane.
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You've studied a lot about lines in your algebra one class,
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but those same ideas connect into geometry as well.
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For example, we know that if two lines have the same slope,
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then those lines are parallel.
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They'll never cross.
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In your slide here,
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you see two parallel lines that are white.
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We have 2x plus two and 2x minus one.
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Both those lines have slope two so they never intersect.
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Now perpendicular lines have what we call
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opposite sign reciprocal slopes.
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So if I wanted to start with the slope two
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and find out something that's perpendicular to that,
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I would do the opposite sign like negative and reciprocal
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so instead of two, it would be 1/2.
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So you see on your slide here a yellow line
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where we have y equals negative 1/2x plus four
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and the negative 1/2 bit, the slope part,
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not the four but the negative 1/2
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is what tells us that that yellow line
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is gonna be perpendicular to both of the white lines.
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Again, parallel lines have the exact same slope.
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Perpendicular lines have opposite sign reciprocal slopes.
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You can also think about
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how the slopes of perpendicular lines
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will multiply to negative one.
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Like for example if I say 2/3,
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the perpendicular slope would be negative 3/2.
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Change the positive to negative and flip the fraction.
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We call that the reciprocal.
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Let's talk also about
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how we can connect coordinate geometry to polygons.
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We can use what we know about the coordinate plane
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to verify our conjectures about polygons,
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similarity, congruence, and right triangles.
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The distance formula will tell us
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that congruent segments are side lengths
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and the slopes of lines can tell us
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if sides are perpendicular or create congruent triangles.
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So let's talk a little bit more about that.
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If we wanna know if two shapes are similar,
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two figures are similar,
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that means that their corresponding
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side lengths are proportional.
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There's some growth factor or maybe a shrinking factor
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that are multiplying each side length by
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to get to the other side length.
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And if it was in the Cartesian coordinate plane,
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I could use the distance formula
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to calculate those distances
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and see am I multiplying everything by the same number,
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like multiplying the smaller distances by two say
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to get to the larger ones
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and so we could check similarity.
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Congruence, of course, is when the side lengths
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are the same.
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That's important as well.
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And then right angles, oftentimes we're asked,
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do these three points create a right triangle?
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And one of the ways we could tell
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is by either finding the distances
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and doing the Pythagorean Theorem
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or finding the slopes and looking to see
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if any of those slopes create perpendicular lines.
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Now you could also look to see
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if you have perpendicular lines
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like it says in your slide using matrices and dot products,
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but many times just calculating the slope
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using the slope formula is a little bit easier.
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We can use coordinate geometry to do proofs,
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in particular you combine your algebra skills
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with your geometry skills to verify a conjecture.
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So here we are asked to classify the quadrilateral ABCD
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formed by the vertices and we have four points.
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Now this graph is already provided for you.
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If you are doing this problem,
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you probably wanna draw a sketch
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and it's okay if your sketch isn't perfect.
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But what you're looking for is to classify a quadrilateral,
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some things you might wanna consider
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are the distances of the side lengths,
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are any of the distances the same
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and you would use the distance formula,
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or the slopes, are any of the slopes the same,
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that would show that you have opposite sides
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or some sides that are parallel,
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that's a key idea of parallelograms and trapezoids,
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those kinds of things.
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Another thing you might wanna look at
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do the slopes show that any of these angles
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are right angles?
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Again, if you have slopes that are opposite sign reciprocals
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like say a slope of two and negative 1/2,
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that would show you that you had some right angles.
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So let's say you are sketching these four vertices by hand,
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not necessarily on graph paper,
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it might look like it could be maybe a rectangle.
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It might look like you might have right angles in there.
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So what you wanna do is find the slopes
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of all four of those sides using the slope formula
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and you'll find that the two opposite sets of slopes
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are perpendicular.
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There's two red lines that both have slopes of negative 1/4,
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the two yellow lines both have slopes of positive three,
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so the opposite sides are parallel.
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That way we know it's some type of parallelogram,
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but those slopes are not giving us right angles.
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They're not opposite sign reciprocals.
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Therefore, I know it's not a rectangle or a square.
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It's just gonna be a parallelogram.
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In isometry or combination of isometries
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An often used to transform a shape in the coordinate plane
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and by transform we mean rotate, reflect,
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or just shift or move.
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So here's an example.
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A coordinate rule shows a transformation.
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So here's an example.
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Every x, y point is getting mapped
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to the x value plus two and the y values minus three.
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Now imagine in your head
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if you had a point or a shape in the coordinate plane,
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if all of your x numbers become those x numbers plus two,
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that's gonna take your shape and move it two to the right
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because adding two is a side-to-side shift
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to the x coordinate.
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We're moving it two to the right.
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And in the y values minus three,
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if you imagine a shape
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and you take all those y values and subtract three,
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that shape will move down.
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It's called an isometry
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'cause we're not changing how big that shape is.
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We're not blowing it up or shrinking it.
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All we're doing is taking that exact same shape
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and shift to get over and down.
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It's called an isometry.
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The second example says all of the coordinates x, y
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become opposite signs, negative x and negative y.
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And what happens is you end up reflecting the shape
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across both axes.
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First you're reflecting across the y-axis
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and then the x-axis
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and those two put together could be called
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either a double reflection or just rotation 180 degrees
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around the origin.
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The last example we have here
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is if you're just changing the sign
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on one of your coordinates
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like if x, y gets mapped to negative x, y.
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Well, if you imagine a shape
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with say like in the first quadrant
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and now all those x values become negative x values,
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it's gonna reflect across the y-axis.
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It's gonna go to coordinate two or quadrant two.
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It's still gonna be the same shape.
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It's still an isometry.
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We're not dilating or shrinking it.
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They're still congruent,
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but it's just we've moved it from one quadrant to the other.
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We can also graph shapes in a 3D coordinate plane
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if we add a third axis called the z-axis.
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And so the way we draw these usually
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is by starting with x and y
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with a right angle as we usually do
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and then we have a diagonal going down to the left z-axis
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like you see on your slide.
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And so each point now doesn't just have x, y coordinates,
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it has x, y, and z.
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And we oftentimes if we're plotting a 3D point like that,
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we'll draw the whole rectangular prism,
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the whole box that creates that
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so you can see in the box how it's moving over side to side,
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it looks like maybe three units.
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It's moving up about two units
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and it's coming towards us, three units.
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So you could see how you would turn a point into a 3D space.