Dilate the rectangle 𝐴𝐵𝐶𝐷 from the origin by a scale factor of one-third, and state the coordinates of the image.
We’ll simply begin by recalling what we actually mean by the origin. The origin is the point with coordinates zero, zero. Now, on our graph, that’s marked with the letter 𝐸. It’s the point where the 𝑥-axes and 𝑦-axes intersect.
Now, we’re looking to dilate the original rectangle 𝐴𝐵𝐶𝐷 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. We’re 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 𝑂𝐵 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 𝐴𝐵𝐶𝐷, the point 𝐵 dash — that’s the image of vertex 𝐵 — will be one diagonal away from the origin. That’s the point with coordinates one, one. And whilst we could do this next for 𝐴, 𝐶, and 𝐷, we actually now know the dimensions of our rectangle. We said it’s one unit wide. So, 𝐴 dash, the image of 𝐴, is one unit horizontally away from 𝐵 dash. And it’s two units high. So, 𝐶 dash, the image of 𝐶, will be two units away from 𝐵 dash. We then complete the diagram as shown.
And so, we’ve dilated the rectangle 𝐴𝐵𝐶𝐷 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 𝐴 dash, the image of 𝐴, is at point two, one; 𝐵 dash is at one, one; 𝐶 dash is at one, three; and 𝐷 dash is at two, three.