### Video Transcript

Dilate triangle ๐ด๐ต๐ถ from the origin by a scale factor of two, and state the coordinates of the image.

Weโre being asked to dilate this triangle โ thatโs sometimes called enlarge the triangle โ from the origin by a scale factor of two. And so we begin by recalling what we mean by the origin. The origin has coordinates zero, zero. Itโs the point of intersection of the ๐ฅ- and ๐ฆ-axes. When we enlarge by a scale factor of two, we ensure that all the dimensions of our shape are multiplied by two. For example, consider the line ๐ต๐ถ. Itโs one, two, three, four, five units in length. This side on our dilated or enlarged triangle will have a length of five times two, which is 10 units.

So how do we perform this enlargement and ensure it ends up in the right place? Well, one thing we can do is add rays to our diagram. These are lines that pass through the center of enlargement โ here thatโs the origin โ and each of our vertices of our triangle. So our first ray passes through the origin and point ๐ด. Our second ray passes through the origin and point ๐ต. And our third passes through the origin and point ๐ถ. Now we know that the vertices of the image, the dilated triangle, must lie on these lines. And so we look at the distance between the origin and each of our vertices and double it.

The distance between the origin and vertex ๐ด is two units. When we multiply this by two, we get four. So vertex ๐ด is transformed onto this point here. Itโs four units away from the origin in the same direction. Now, ๐ต is a little trickier. So what we could do is count the horizontal and vertical distance from the origin to ๐ต. We move one unit right and two units up. By enlarging by a scale factor of two, weโd end up moving two units right and four units up. So the transformed vertex ๐ต is as shown. Itโs at the point two, four.

Weโll repeat this process for ๐ถ. This time, to get from the origin โ thatโs the center โ to point ๐ถ, we move one unit right and three units down. To get to vertex ๐ถ on our enlarged or dilated triangle, weโre going to double these measurements. And we move two units right and six units down. We can therefore see that the image of ๐ถ is as shown. To complete the dilated triangle or the image of ๐ด๐ต๐ถ, we add in the sides. And what we could do is check that the length between the enlarged side ๐ต๐ถ is 10 units as we said we required. If we count, we see that it is indeed 10 units. And so the image of ๐ด๐ต๐ถ is as shown.

The second part of this question asks us to state its coordinates. Well, we have the image of ๐ด, which is at negative four on the ๐ฅ-axis. That has coordinates negative four, zero. The image of ๐ต is two, four, and the image of ๐ถ is two, negative six.