### Video Transcript

Dilate the rectangle ๐ด๐ต๐ถ๐ท from the origin by a scale factor of one-third, and state the coordinates of the image.

Weโll simply begin by recalling what we actually mean by the origin. The origin is the point with coordinates zero, zero. Now, on our graph, thatโs marked with the letter ๐ธ. Itโs the point where the ๐ฅ-axes and ๐ฆ-axes intersect.

Now, weโre looking to dilate the original rectangle ๐ด๐ต๐ถ๐ท about this point by a scale factor of one-third. So, what does that actually mean? Well, firstly, it means that all dimensions of the rectangle itself will be a third of the original dimensions. Currently, the rectangle is three units by six units. A third of three is one, and a third of six is two. So, we know the dimensions of our new rectangle, the dilated rectangle, will be one unit by two units.

But we need to figure out where this rectangle is going to be. And this is where the center comes into play. Weโre told to dilate the rectangle about the origin. And so, we look at the distance of each vertex from that center, from the origin, and we multiply those by one-third. Now, for more complicated lengths, we can consider the horizontal and vertical distances independently. In this case though, the line ๐๐ต passes exactly through the diagonal of three squares.

We said that one-third of three is one. So, for our image, the dilation of the rectangle ๐ด๐ต๐ถ๐ท, the point ๐ต dash โ thatโs the image of vertex ๐ต โ will be one diagonal away from the origin. Thatโs the point with coordinates one, one. And whilst we could do this next for ๐ด, ๐ถ, and ๐ท, we actually now know the dimensions of our rectangle. We said itโs one unit wide. So, ๐ด dash, the image of ๐ด, is one unit horizontally away from ๐ต dash. And itโs two units high. So, ๐ถ dash, the image of ๐ถ, will be two units away from ๐ต dash. We then complete the diagram as shown.

And so, weโve dilated the rectangle ๐ด๐ต๐ถ๐ท about the origin by a scale factor of one-third. Next, we need to state the coordinates of the vertices of this image. We can see ๐ด dash, the image of ๐ด, is at point two, one; ๐ต dash is at one, one; ๐ถ dash is at one, three; and ๐ท dash is at two, three.