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Dilating shapes: shrinking

Draw the image of the triangle under a dilation with scale factor 1/4 about the center of dilation on the coordinate plane.

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Video transcript

- [Instructor] We're told draw the image of triangle ABC under a dilation whose center is P and scale factor is 1/4. And what we see here is the widget on Khan Academy where we can do that. So we have this figure, this triangle ABC, A, B, C, right over here, and what we wanna do is dilate it, so that means scaling it up or down, and the center of that dilation is this point P. So one way to think about it is let's think about the distance between point P and each of these points, and we wanna scale it by 1/4. So the distance is going to be 1/4 of what it was before. So, for example, this point right over here, if we just even look diagonally from P to A, we can see that we are crossing one square, two squares, three squares, four squares. So if we have a scale factor of 1/4, instead of crossing four squares diagonally, we would only cross one square diagonally. So I'll put the corresponding point to A right over there. Now, what about for point C? It's not quite as obvious, but one way we could think about it is we can think about how far are we going horizontally from P to C, and then how far do we go vertically? So horizontally, we're going one, two, three, four, five, six, seven, eight of these units, and then vertically we're going one, two, three, four. So we're going to the left eight and up four. Now, if we have a scale factor of 1/4, we just multiply each of those by 1/4. So instead of going to the left eight, we would go to the left two. Eight times 1/4 is two. Instead of going up four, we would go up one. So this would be the corresponding point to point C. And then we'll do the same thing for point B. When we go from P to B, we're going one, two, three, four, five, six, seven, eight up, and we're going four to the left. So if we have a scale factor of 1/4, instead of going eight up, we'll go two up, and instead of going four to the left, we'll go one to the left. So there you have it. We have just dilated triangle ABC around point P with a scale factor of 1/4, and we are done.