Graph the image of each triangle after a dilation with the given scale factor. Then verify that

Graph the image of each triangle after a dilation with the...

Key Concepts

Dilation

Dilation is a transformation that produces an image that is the same shape as the original while changing its size. In a dilation, every point on the original figure is moved along a straight line connecting it to a fixed center of dilation, and the distances from the center are multiplied by a scale factor.

Scale Factor

The scale factor in a dilation represents the ratio of distances between the corresponding points of the original figure and its image. A scale factor of 1/2 means that each distance from the center of dilation is halved, resulting in an image that is a reduced but proportional version of the original.

Similarity

Similarity is a property of figures that have the same shape regardless of their size. Two figures are similar if all corresponding angles are congruent and the corresponding side lengths are in proportion, which is precisely what is achieved through dilation.

Coordinate Geometry Transformation

In coordinate geometry, transformations like dilation are applied to the coordinates of the key points of a figure. By multiplying the coordinates by the scale factor, you obtain the coordinates of the image, allowing for a precise graphing of the transformed figure.

Transcript

00:03 Hi, we're looking at question 16 that asks us to graph the new triangle after a dilation of one -half.

00:12 Our original points, we had m, which was 0 .4.

00:16 So when we multiply that by one -half, our scale factor, our m -prime is going to be 0 -2.

00:27 Then we had n, which was 4 -2.

00:34 And when you multiply that by 1 -half, n -prime will be 2 -1, because remember, multiply both parts by one half.

00:42 And then p is two negative two.

00:48 And when you multiply that by one half, you get one negative two.

00:54 So i'm doing a quick sketch of the graph because i don't have graph paper on here.

01:00 If you have graph paper, certainly use that.

01:03 If you don't have graph paper, you can always do a sketch like i'm doing.

01:07 I'm just going to put the few points that we need on the axes.

01:10 So these would be by ones.

01:12 So we have negative one here, negative one here, one there, and one there.

01:18 So 0, 2 would be right here.

01:23 That's my m.

01:25 2 .1 would be right here.

01:28 That's my n.

01:30 And i can connect those.

01:32 And then 1 negative 2, that's my p, would be right here.

01:38 And these would connect.

01:40 So there's my sketch of my triangle.

01:45 And then it wants us to verify that the triangles are similar.

01:50 So to do that, we're going to actually have to, since we can't get any angles out of these, we're going to have to actually calculate the lengths of the various sides.

02:01 So to do that, we'll go back to our distance formula.

02:04 And i'll explain this first one out in detail, but after that i'll just kind of go through it.

02:09 So you're going to take the x.

02:11 Coordinate, it really doesn't matter whether you start with m or n.

02:14 I usually start with the second one and then go to the first one, but you can do it the other way.

02:19 It's not a problem.

02:20 So the four is the x coordinate of the n and then the zero is the x coordinate of the m.

02:26 And then when you do y, you have to go in the same order.

02:29 So since i started with n before, i'm going to start with n again.

02:32 So it's two minus four all squared.

02:38 So then i'm going to simplify that.

02:43 And four minus zero is 4 and then i square it is 16 and 2 minus 4 is negative 2 and i square that and i get 4.

02:53 So that's equal to square root of 20 and then we should simplify that.

02:59 So square to 20 is the same as square root of 4 times square to 5 and square to 4 is 2.

03:04 So it's 2 square root 5.

03:07 And now we want to do m prime n prime the same way.

03:18 So i would do 2 minus 0...

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