WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For the next lesson, we are going to go over mappings.
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Mappings are transformations of a pre-image to another congruent or similar image.
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When you have an image, and it has either a congruent or a similar relationship with another image,
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then that is transformations, which is also mappings.
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The different types of transformations are rotation, translation, reflection, and dilation--four types of transformations.
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The first one, **rotation**, is when you take the pre-image (the pre-image is the original, the initial, the first image), and it rotates.
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So, you turn it to make the second image; it is just rotating or turning.
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**Translation** is when you take the image and you slide it, so it just moves; that is it.
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It doesn't rotate; it doesn't do anything but just move--a slide or a glide.
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**Reflection** is when you flip the image: you have two images, and they are just reflections of each other.
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And **dilation** is when you enlarge or reduce the image.
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Again, transformations are when you perform one of these four to a pre-image to create another image that is either congruent or similar.
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Here are just some image: with rotation, you take this image (this is the pre-image), and to make this image, all I did was rotated it--just turned.
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It is the same image, and it just rotated.
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Translation: again, this is the pre-image, and it just slides or glides--just moves.
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It stays the same; it just moves to a different location, a different place right here.
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Reflection is, again, like a mirror reflection; they are reflections of each other.
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And dilation is when an image gets larger or smaller; this is the same shape, but different size.
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It just gets bigger, or it gets smaller; but it has to be the same shape.
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And if two images have the same shape, but a different size, then we know that they are similar.
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With dilation, it will be similar images; so then, the other three (rotation, translation, and reflection) are all congruence transformations,
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because when you perform these transformations, they don't change; they are still congruent in size and shape.
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Nothing changes; it is just the way you position it, or the way you rotate or reflect or translate the image; it is just going to stay the same.
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And that is called an isometry; an isometry is a transformation that maps every segment to a congruent segment.
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Again, when you either rotate, translate, or reflect, the images are congruent; they are the same.
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Describe the transformation that occurred in the mappings.
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Here, we want to know what happened with this image to get this image.
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All that this did was to turn; so from this to this, it just turned a certain angle amount; and so, this is rotation, because it just rotated.
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This right here, from this image to this image, the pre-image to the image, looks like a reflection; it looks like it is looking in a mirror.
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It reflects, so this is reflection.
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And then, for these two, see how one is bigger than the other.
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So, even though it kind of looks like reflection, it can't be, because reflection has to be exactly the same.
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It has to line up exactly the same way and be the same size; they have to be congruent.
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But here, because this image and this image are different sizes, but the same shape, this has to be dilation.
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Think of dilation as...when something dilates, it gets bigger; so it is getting bigger or getting smaller.
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The next example: Determine if the transformation is an isometry.
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Remember: an isometry is when you have two images, and the pre-image and the image are congruent; that is for rotation, reflection, and translation.
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We just want to see if these two are congruent.
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Now, to determine if two triangles are congruent, remember: we have those theorems and postulate,
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where it says Angle-Side-Angle (they are corresponding parts), Side-Angle-Side, Side-Angle-Angle, and Side-Side-Side.
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Those are the different congruence theorems and postulate.
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We want to see if this pair of triangles applies to any of those.
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Now, here I see that an angle is congruent here, and a side, and they are corresponding parts.
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Now, for this one, angle B is corresponding with angle E; this one is given, and this one is not.
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And angle C is corresponding with angle F, but this one is not given, and this one is.
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I want to find the measure of this angle, and I can do that by taking these two and subtracting it from 180; so it is 180 - (105 + 40).
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This right here is 145; so if you subtract this from 180, you will get 35 degrees.
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The measure of angle C is 35 degrees.
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Now, the measure of angle E is going to be 180 - (35 + 40); now, we don't have to solve for that, because we know that this is 35;
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this angle is congruent to this angle; and of course, that means that this angle has to be congruent to this angle.
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So, I have Angle-Side-Angle, because this pair of angles is congruent; their sides are congruent; and the angles are congruent.
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So, because of this, these two triangles are congruent, and therefore, this is an isometry; so it is "yes."
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The next one: Show that triangle ABC and triangle DEF are an isometry (so it is the same type of problem).
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Now, for this, we have the coordinates of each vertex for each triangle.
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So, we can find the measure, or the length, of each side.
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I can just find the measure of that side with the length of that side and compare them and see if they are congruent.
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I have to use the distance formula: the distance formula is (x₂ - x₁)² + (y₂ - y₁)².
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Remember: this means the second x; so it is the second x, minus the first x, and the second y, minus the first y.
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A is (-6,1); B is (-4,6); and C is (-2,3); then, D is (1,-1), E is (3,4); and F is (5,1); find the distance of AB.
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So, AB is, let's see, (-4 + 6)², and then (6 - 1)².
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I have that this is 2 squared, plus 5 squared, which is 4 + 25, which is √29.
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And then, let's do DE: DE is (3 - 1)² + (4 + 1)²; and it is plus because it is 4 - -1.
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So, this is 2 squared, plus 5 squared, the same as that; so it is the square root of 29.
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I know that AB is congruent to DE; now, BC is (-2 + 4)² + (3 - 6)²;
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so, this is 2 squared, plus -3 squared, which is 4 +...this is 9; that is √13.
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And then, what is corresponding with BC? EF.
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EF is (5 - 3)² + (1 - 4)²: 2 squared plus -3 squared is the same, √13; so those two are the same.
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And then, AC (I am running out of room here) is (-2 + 6)² + (3 - 1)².
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That is 4² + 2²; this is 16 + 4, is 20; √20...
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And then, AC and DF...DF is (5 - 1)² + (1 + 1)², so this is 4² + 2²,
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16 + 4; that is 20, so that is √20; so then, these two are the same.
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So, by the Side-Side-Side Congruence Theorem, they are congruent, which means that it is an isometry.
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That is it for this lesson; thank you for watching Educator.com.