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Finding measures using rigid transformations

Finding measures after a rigid transformation, like a reflection, is pretty simple! Since the shape and size stay the same, the lengths of corresponding sides and angle measures remain unchanged. Area and perimeter depend on the side lengths, so they stay the same too. So, if we know the measures of the original figure, we can use those same measures for the transformed figure.

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Video transcript

- [Instructor] We are told that triangle ABC, which is right over here, is reflected across line l, so it's reflected across line l right over here, to get to triangle A prime B prime C prime. Fair enough. So based on that, they're going to ask us some questions. And I encourage you to pause this video and see if you can figure out the answers to these questions on your own before I work through them. So the first question they say is well, what's A prime C prime? This is really what's the length of segment A prime C prime? So they want the length of this right over here. How do we figure that out? Well, the key realization here is a reflection is a rigid transformation. Rigid transformation, which is a very fancy word. But it's really just saying that it's a transformation where the length between corresponding points don't change. If we're talking about a shape like a triangle, the angle measures won't change, the perimeter won't change, and the area won't change. So we're gonna use the fact that the length between corresponding points won't change. So the length between A prime and C prime is gonna be the same as the length between A and C. So A prime C prime is going to be equal to AC, which is equal to they tell us right over there. That's this corresponding side of the triangle. That has a length of three. So we answered the first question. And maybe that gave you a good clue. And so I encourage you to keep pausing the video when you feel like you can have a go at it. Alright, the next question is what is the measure of angle B prime? So that's this angle right over here. And we're gonna use the exact same property. Angle B prime corresponds to angle B. It underwent a rigid transformation of a reflection. This would also be true if we had a translation, or if we had a rotation. And so right over here, the measure of angle B prime would be the same as the measure of angle B. But what is that going to be equal to? Well, we can use the fact that if we call that measure, let's just call that X. X plus 53 degrees, we'll do it all in degrees, plus 90 degrees, this right angle here. Well, the sum of the interior angles of a triangle add up to 180 degrees. And so what do we have? We could subtract, let's see, 53 plus 90 is X, plus 143 degrees is equal to 180 degrees. And so subtract 143 degrees from both sides. You will get X is equal to, let's see, 180 minus 40 would be 40. 80 minus 43 would be 37 degrees. X is equal to 37 degrees, so that is 37 degrees. If that's 37 degrees, then this is also going to be 37 degrees. Next, they ask us what is the area of triangle ABC? ABC. Well, it's gonna have the same area as A prime B prime C prime. And so a couple of ways we could think about it. We could try to find the area of A prime B prime C prime based on the fact that we already know that this length is three and this is a right triangle. Or we can use the fact that this length right over here, four, from A prime to B prime is gonna be the same thing as this length right over here, from A prime to B prime, which is four. And so the area of this triangle, especially this is a right triangle, it's quite straightforward, it's the base times the height times 1/2. So this area is gonna be 1/2 times the base, four, times the height, three, which is equal to 1/2 of 12, which is equal to six square units. And then last but not least, what's the perimeter of triangle A prime B prime C prime? Well, here we just used the Pythagorean Theorem to figure out the length of this hypotenuse. And we know that this is a length of three based on the whole rigid transformation and lengths are preserved. And so you might immediately recognize that if you have a right triangle where one side is three and the other side is four, that the hypotenuse is five. Three four five triangles. Or you could just the Pythagorean Theorem. You say three squared plus four squared, four squared is equal to let's just say the hypotenuse squared. Well, three squared plus four squared, that's nine plus 16. 25 is equal to the hypotenuse squared. And so the hypotenuse right over here will be equal to five. And so they're not asking us the length of the hypotenuse. They wanna know the perimeter. So it's gonna be four plus three plus five, which is equal to 12. The perimeter of either of those triangles, because it's just one's the image of the other under a rigid transformation. They're gonna have the same perimeter, the same area. The perimeter of either of the triangles is 12. The area of either of the triangles is six. And we're done.