Finding measures using rigid transformations

We are told that triangle ABC, which is right over here, is reflected across line L. So it's reflected across the 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.

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, 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 three.

So we answered the first question, and maybe that gave you a good clue. I encourage you to keep pausing the video when you feel like you can have a go at it.

All right, the next question is, what is the measure of angle B prime? So that's this angle right over here, and we're going to use the exact same property. The measure of 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. 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 adds up to 180 degrees. 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 minus 143 would be 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? Well, it's going to have the same area as A prime B prime C prime.

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, 4, from A prime to B prime, is going to be the same thing as this length right over here from A to B, which is 4.

The area of this triangle, especially since this is a right triangle, is quite straightforward. It's the base times the height times one-half. So this area is going to be one-half times the base 4 times the height 3, which is equal to half of 12, which is equal to 6 square units.

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 we know that this is a length of three based on the whole rigid transformation and lengths are preserved. So you might immediately recognize that if you have a right triangle where one side is three and one other side is four, that the hypotenuse is five. Three, four, five triangles!

Or you can just use the Pythagorean theorem. You say three squared plus four squared is equal to the hypotenuse squared. Well, three squared plus four squared, that's nine plus sixteen. Twenty-five 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 want to know the perimeter. So it's going to be four plus three plus five, which is equal to twelve.

The perimeter of either of those triangles, because it's just one's the image of the other under a rigid transformation, they're going to have the same perimeter, the same area. The perimeter of either of the triangles is twelve, the area of either of the triangles is six, and we're done.