Dilate triangle 𝐴𝐵𝐶 from the origin by a scale factor two, and state the coordinates of the image.
We’re going to dilate or enlarge our triangle, and the centre that we’re going to use is the origin. That’s the point whose coordinates are zero, zero, as shown Now, we’re going to use a scale factor of two. That means each dimension of our triangle will be twice the size. It will double. But it’s important that each coordinate is double the distance, twice the distance, away from the centre of enlargement. And so, to achieve the dilation in the correct place, we might look to draw some rays.
Each ray passes through the origin and a vertex of the triangle and we could, if we so wished, measure the distance of each vertex from the origin and then double that distance along that ray. So, for example, let’s take vertex 𝐶. It’s two units away from the origin horizontally. Which means the image of 𝐶 will be four units away from the origin horizontally, taking us to a point with coordinates negative four, zero. And I’ve written 𝐶 dash to show that this is the image of 𝐶. It’s the dilation.
But what about point 𝐴? Well, to get from zero to point 𝐴, we move five units horizontally in the negative direction and then two units vertically upwards. And so, this is a little bit tricky because when we double these distances, we end up moving 10 units horizontally and four units up. That must take us to the point with cartesian coordinates negative 10, four. And unfortunately, since this’s come off of our grid, we can only estimate its location.
Let’s repeat this with point 𝐵. This time to move from the origin to point 𝐵, we move three units horizontally in the negative direction and then four units up. And so, to get from zero to the image of 𝐵, we’re going to double these distances. We’re going to move six units left and eight units up. That takes us to the point with coordinates negative six, eight. And so, we add the image of 𝐴𝐵𝐶, its dilation, as shown. And if we were to measure each of the sides of this triangle, we would see that each side is now double the length of the original triangle. The coordinates of the image of 𝐴𝐵𝐶 are negative 10, four; negative six, eight; and negative four, zero.