Congruence and similarity
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The graph below contains triangle ABC and the point P. Draw the image of triangle ABC under a dilation whose center is P and a scale factor of 2. So essentially, we want to scale this so that every point is going to be twice as far away from P. So for example, B right over here has the same y-coordinate as P, but its x-coordinate is three more. So we want to be twice as far. So if this maps to point B, we just want to go twice as far. So we're at 3 away, we want to go 6 away. So point P's x-coordinate is at 3, now we're at 9. Likewise, point C is 3 below P. Well we want to go twice as far, so we'll go 3 more. And point A is 4 above P. Well we want to go 4 more. We want to go twice as far-- one, two, three, four. And we get right over there. Then they ask us, what are the lengths of side AB and its image? AB right over here, let's see, we might have to apply the distance formula. Let's see, it's the base right over here. The change in x between the two is 3 and the change in y is 4, so this is actually a 3, 4, 5 right triangle. 3 squared plus 4 squared is equal to 5 squared. So AB is 5 units long. Essentially just using the Pythagorean theorem to figure that out. And its image, well it's image should be twice as long. And let's see whether that actually is the case. So this is a base right over here that's of length 6. This has a height, or this change in y, I could say. Because I'm really just trying to figure out this length, which is the hypotenuse of this right triangle. I don't have my drawing tool, so I apologize. But this height right here is 8. So 8 squared is 64, plus 6 squared is 36, that's 100, which is 10 squared. So notice, our scale factor of 2, the corresponding side got twice as long. Each of these points got twice as far away from our center of dilation.