Video Transcript
Find the images of the vertices of the quadrilateral 𝐴𝐵𝐶𝐷 after a dilation with center 𝐶 by a scale factor of nine-tenths.
Let’s think about what it means for the quadrilateral to be dilated by a scale factor of nine-tenths about center 𝐶. It means that all the vertices of the quadrilateral 𝐴𝐵𝐶𝐷 will be nine-tenths of the distance away from 𝐶 as they were originally. Now, currently, 𝐶 is zero units away from itself. Nine-tenths of zero is zero. So, the image of vertex 𝐶, 𝐶 dash, will be in the same place. That’s got coordinates of negative five, negative four.
We’ll now consider the length of the line 𝐵𝐶. It’s currently three units in length. Since the scale factor is nine-tenths, we know that the image of vertex 𝐵, 𝐵 dash, will be nine-tenths or three units away from 𝐶. Well, remember, we can calculate this by working out nine-tenths times three or nine-tenths times three over one. We multiply two fractions by multiplying their numerators and individually multiplying their denominators. So, nine-tenths times three over one is twenty-seven tenths. And so, we can say that the image of vertex 𝐵 will be twenty-seven tenths away from 𝐶.
Now, of course, that’s only in the horizontal direction. So, it follows that the 𝑦-coordinate of 𝐵 will remain unchanged. It’s negative four. Since the 𝑥-coordinate of 𝐶 is negative five, we can say that the 𝑥-coordinate of 𝐵 dash, the image of 𝐵, will be negative five minus twenty-seven tenths. Once again, we make negative five be equal to negative five over one. And then, we’re going to create a common denominator. Now, here, we can create a common denominator of 10 by multiplying the numerator and denominator of our first fraction by 10. Then, once the denominators are equal, we simply subtract their numerators. And we get negative seventy-seven tenths. And so, we found the image of 𝐵. It’s 𝐵 dash at negative seventy-seven tenths, negative four.
Let’s repeat this process, next looking at the length of the line 𝐶𝐷. It’s currently four units in length. This means that the image of vertex 𝐷, 𝐷 dash, will be nine-tenths of four units away from 𝐶. This time that’s in the vertical direction. Of course, this is nine-tenths times four over one, which is thirty-six tenths or eighteen-fifths. Now, we said that it is only in the vertical direction. So, the 𝑥-coordinate of the image of 𝐷, 𝐷 dash, will still be negative five.
The 𝑦-coordinate will be found by subtracting eighteen-fifths from the 𝑦-coordinate of 𝐶. That’s negative four. Once again, we write negative four as negative four over one. Then, we multiply the numerator and denominator of this fraction by five to create equivalent denominators. Negative twenty-fifths minus eighteen-fifths is negative thirty-eight fifths. And so, we have the vertex 𝐷 dash. It’s the image of the vertex 𝐷, and it lies at negative five, negative thirty-eight fifths.
And what about 𝐴 dash? That’s the image of the vertex 𝐴. If we were sketching the image of 𝐴𝐵𝐶𝐷 out, we could look at the distance that 𝐴 dash is away from 𝐵 dash. In this case though, let’s look at the distance that 𝐴 is from 𝐶. It’s currently three units in the vertical direction and three units in the horizontal direction away from 𝐶. On the image of 𝐴𝐵𝐶𝐷, 𝐴 dash will be nine-tenths of three units both horizontally and vertically away from the center 𝐶. We saw before that that was twenty-seven tenths away. And we already worked out that twenty-seven tenths in the negative horizontal direction from negative five was negative seventy-seven tenths.
To find the 𝑦-coordinate, we’re going to subtract twenty-seven tenths from the 𝑦-coordinate of 𝐶. That’s negative four. Once again, we write negative four as negative four over one, and then we create a common denominator of 10. Negative 40 minus 27 is negative 67. And so, we find the 𝑦-coordinate of the image of 𝐴 to be negative sixty-seven tenths.
And so, we found the coordinates of the images of the vertices of the quadrilateral 𝐴𝐵𝐶𝐷 after the given dilation. 𝐴 dash is negative seventy-seven tenths, negative sixty-seven tenths. 𝐵 dash is negative seventy-seven tenths, negative four. 𝐶 dash is negative five, negative four. And 𝐷 dash is negative five, negative thirty-eight fifths.
And, in fact, we can perform a little check on our answers. We would expect 𝐴 dash and 𝐵 dash to be vertically above one another. This means their 𝑥-coordinates should be the same, as they are. Similarly, as in the original shape, 𝐶 and 𝐷 are above one another. So, 𝐶 dash and 𝐷 dash should be above one another, and their 𝑥-coordinates are the same. Finally, we see in our original shape, 𝐵 and 𝐶 are horizontally in line. So, in the dilated shape, we would expect 𝐵 dash and 𝐶 dash to be horizontally in line, meaning their 𝑦-coordinates should be the same, and they are.
