Question Video: Dilating Shapes From a Point by Positive Scale Factors Mathematics • 8th Grade

Video Transcript

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

We’re being asked to dilate this triangle — that’s sometimes called enlarge the triangle — 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. It’s the point of intersection of the 𝑥- and 𝑦-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 𝐵𝐶. It’s 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 — here that’s the origin — and each of our vertices of our triangle. So our first ray passes through the origin and point 𝐴. Our second ray passes through the origin and point 𝐵. And our third passes through the origin and point 𝐶. 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 𝐴 is two units. When we multiply this by two, we get four. So vertex 𝐴 is transformed onto this point here. It’s four units away from the origin in the same direction. Now, 𝐵 is a little trickier. So what we could do is count the horizontal and vertical distance from the origin to 𝐵. We move one unit right and two units up. By enlarging by a scale factor of two, we’d end up moving two units right and four units up. So the transformed vertex 𝐵 is as shown. It’s at the point two, four.

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

The second part of this question asks us to state its coordinates. Well, we have the image of 𝐴, which is at negative four on the 𝑥-axis. That has coordinates negative four, zero. The image of 𝐵 is two, four, and the image of 𝐶 is two, negative six.