Dilations in the Coordinate Plane 8 G A 3

hi welcome to the magic of math where we mastermath one video at time my video lesson today is on dilations our objectives today are that you will identify dilation and that you will dilate figures in the coordinate plane here's the question I'd like you thinking about today as proceed through the lesson how can you tell when dilation is reduction or an enlargement let's begin by reviewing vocabulary transformation describes how two-dimensional figure moves on coordinate plane transformation is change in location by sliding turning flipping or changing size so we're going to talk about changing size today which is dilation figure changes size not shape by becoming smaller or larger about fixed Point called the center of dilation dilated figure increases or decreases specific amount the scale factor dilated image is always similar to the original figure center of dilation is fixed point in plane that all points are expanded or shrunk by specific Factor the point can be inside or outside of the figure similar figures are two figures that have the same shape corresponding angles will be congruent and corresponding sides will be proportional and our last is an image this is the figure after transformation the image is labeled using this apostrophe and in math is read as Prime so for example triangle ABC transforms to Triangle prime Prime Prime let's begin by understanding that dilations exist in our real world here are pair of highs during an eye exam an unoptometrist May dilate your eyes to enlarge your pupils so they enlarge that black pupil in your eye so that they can better examine your eye and see all the way back into your eye so they're enlarging your pupil or the word they use is dilate so in math dilation can be an enlargement and then we have that each letter here in the word dilation is getting bigger with respect to the center of dilation so if we look at these lines they form point here and they're going the same distance and each letter is expanding you can also have dilation that's reduction in size so reduction is one where everything gets smaller but proportionally but still with respect to the center of dilation so even when we get to the coordinate plane if you take your center of dilation and draw lines out it should hit the vertices of your figures an enlargement has scale factor so dilated image is similar to its original figure an enlargement is increased by scale factor that is greater than one understanding that if you multiply anything by one it gets itself so it's not any bigger and in the word scale factor factor is something that you multiply something by so here we have dilate with scale factor of two so that means this triangle is going to increase in size we're going to dilate it so it's larger than itself now but it will be similar figure so all sides will increase by the same scale factor so we're going to take our scale factor and we're going to multiply all sides by the same number so now my new triangle is going to be 3 times 2 the corresponding side will be 6. 4 times 2 that corresponding side will be 8. so we know that corresponding sides are proportional in similar figures three over six simplifies to one-half four over eight simplifies to one-half so we have created dilation that is an enlargement with scale factor of two scale factor can also create smaller figure which we call reduction dilated image is always similar to its original figure reduction is decreased by scale factor greater than zero and less than one meaning it's going to be decimal or fraction so here we're going to take this triangle and we're going to dilate it with scale factor of one-half it means we're going to create triangle that is similar to this triangle but smaller we're going to call that reduction dilation that's reduction so we're going to multiply every side in this triangle by one half to Curry the image eight times one-half is 4. 14 times one-half is seven once again can check my ratios of corresponding sides 8 over 4 is 2. 14 over 7 is 2. so there we have created reduction that creates similar figure because corresponding sides are proportional we can also dilate on coordinate plane so have this figure triangle ABC in the coordinate plane and here's our instructions to dilate figure in the coordinate plane we're going to multiply each coordinate of each vertices by the scale factor graph the image and label the vertices vertices so first thing I'm going to do is identify that my instructions here are dilate triangle ABC using scale factor of 3. so we're going to create in this coordinate plane larger image we know it's going to be an enlargement because my scale factor is greater than 1. we know it's also going to be similar figure to this green triangle ABC so first want to identify the vertices of triangle ABC so we want negative 1 negative 2 negative 3 negative 1. so negative 3 negative 1 is negative 1 negative one and negative 1 positive 2. so now that I've identified the vertices of triangle ABC I'm going to multiply each of these by the scale factor of 3. so each coordinate gets multiplied by the scale factor so prime becomes negative 9 negative three negative 3 times 3 is negative nine negative 1 times 3 is negative 3. now let's do Prime 3 times negative 1 is negative 3 negative 3. and Prime negative 1 times 3 negative 3 and 2 times 3 is 6. so now that have done that I'm ready to graph the image prime will be at negative 9 negative three Prime negative three negative 3 and Prime negative three positive 6. so label my vertices and there you have your enlarged dilation and here at the center of the dilation is the origin so if take the origin and extend out my lines if I've done it correctly will pass through so see this line goes through this center of dilation which is the origin vertices vertices center of dilation vertices vertices center of dilation vertices vertices so if you've done it correctly all vertices will line up with the center of dilation now it's your turn would like you to graph the dilation of triangle ABC using scale factor of 2. please pause the video now and come back when you're ready welcome back so again we are going to dilate this original triangle ABC coordinate A's the coordinates for is knit one negative two negative 3 negative two one two three four two for and then positive three one two three four for our scale factors two so we're going to multiply each coordinate of these three vertices by two prime becomes negative three times two is negative six negative two times two is negative four times two for Prime would be eight two times two is four Prime 3 times 2 is 6 negative 4 times 2 is negative 8. let's go and graph our image so negative 6 4 1 2 3 4 5 6 and down four one two three four for prime we're gonna go over eight and up four for Prime then over six and down eight for Prime so notice that we have an enlargement this time our scale factor is greater than two your turn would like you to graph the dilation of triangle ABC with scale factor of one-half please pause the video now and come back when you're ready welcome back here's our solution so first I'm going to identify my three vertices so have negative eight negative 2 for is positive 2 4 and is negative 6 negative 4. so identifying that we're doing reduction because my scale factor is less than one multiplying each vertices by one-half so negative eight times one-half is negative four negative two times one-half is negative one for Prime two times one half is one four times one-half is two and for Prime six times one half is three and negative four times one-half is negative two now let's graph our image so our first vertices is negative four negative 1 then is 1 2 Prime and Prime negative 3 negative 2. and again you can tell that we have reduction and if we went from the center the center of dilation our origin and Drew lines out they would pass through both vertices proving that we have correctly dilated our figure with scale factor of one-half thank you for joining me today at the magic of math where we continue to master math one video at time hope you'll come back soon and please have great day