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CCSS.Math: ,

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 set 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 transformations which is a very fancy word but it's really just saying that it's a transformation where the length 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 going to use the fact that the length between corresponding points won't change so the length between a prime and C prime is going to 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 3 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 B 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 be 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 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 get X is equal to let's see 180 80 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 3 and this is a right triangle or we can use the fact that this length right over here for 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 4 and so the area of this triangle especially the right triangle is quite straightforward it's the base times the height times 1/2 so this area it's gonna be 1/2 times the base 4 times the height 3 which is equal to 1/2 of 12 which is equal to 6 square units and then last but not least what's the perimeter of triangle a prime B prime C prime well here we just use the Pythagorean theorem to figure out the length of this hypotenuse and so and we knew we know that this is a length of 3 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 3 and one other side is 4 that the hypotenuse is 5 3 4 5 triangles or it could just use the Pythagorean theorem you say 3 squared plus 4 squared 4 squared is equal to let's just say the hypotenuse the hypotenuse squared well 3 squared plus 4 squared that's 9 plus 16 25 is equal to the News 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 want to know the perimeter so it's gonna be four plus three plus five which is equal to twelve the perimeter of either of those triangles because it's just ones 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 the triangles is six and we're done