So problem three asks us about was the minimum height of a mirror that's needed. He needed to see the complete image of the person who is 178 centimeters tall so we can ask this question in a number of please himself, saying, You know, you have the minimum and kind of a mirror, uh given, given the object length going to say, Oh, or we can say the maximum object length given a mirror, I've specified length. So if you answer the question for one, you'll be answering the question for another, because when a question we're asked based on the given object length, what's the minimum mirror needed to see the complete image of this object? However, if you extend this object given a mirror, we can extend this object down to see how much of this mirror, or how much of this object how much of the object can be fit within this mirror image before it cannot be fit anymore. That would be the minimum mirror that would give you the minimum Merrill length given that length of for the object. So just toe drew some review. Um, we know that for flat mirrors you go is gonna have a equal image on the other side of the mirror that has a magnitude off plus one. So in order for you to see um, yourself in the mirror, you're gonna have to have, like, bounce off from you, hit the mirror and then return back to your eyes. That's what's gonna have to happen for you to serious self in the mirror and nor to, um, from this image. That reflective ray is like a It's in virtual Ray, it seems to, it appears to um, to come from an image that's farther back in the mirror, and that's what makes you see yourself in the mirror. Now what also you have to do in order to determine your height. Length is, too. Send a ray of light from you to really any point on the mirror and using the law of reflection, you flip that ray of light and then connect that virtually and then you get your intersection here. Now what I want to do is I want to do this. When I want to do is I want to extend this object to answer the question on the right. What's the maximum object length for a minimum or try for a given mirror. So we have this mirror that we've constructed. And let's say that our object was to be extended below this principal axis. It would be a point. Um, while we're building this object length, then a ray of light bouncing from actually me race. This first, a ray of light balancing from this lowest point of the figure reaches the edge of the mirror and only the edge which reaches your eyes. And the reason why we do it like this, because light from eat flight that's bounced off of the object or you in this case has to go toward the mirror and then reflect based on love, reflection to your eyes. So we want to have a ray of light that reaches to your eyes here to the top, and that is coming from the edge of the mirror. The maximum. Finally, the extreme case seasoning the extreme case in which, after this point, there is no reflection of rays from bouncing from being bounced from your body. So what we want to do is we want to be able to determine the vertical distance um, of this ray being traveled and both one goes to a mirror. And then when it goes to your eyes, then from there, um, you'll see what happens here. You'll see what we're able to come up with. So that's gonna come. Let's come up with these triangles. Say, this is the ray of light that features the, uh, edge, the very edge of a mirror. This is the This represents the maximum object that we can have forgiven mirror. Um, and we know this is a right triangle, since that's based off for construction here. Same thing for this triangle. Um, so we have what we know is that based on love reflection, uh, these angles have to be the same. But also, since this is a light traveling here to the mirror and back, the horizontal displacement is the same for both raise or for the same, right. Rather. So we have an angle aside, an angle that are the same for both of these triangles, making them, uh, congruent. We have We have a triangle that lets us know that this I hear this in the vehicle to the side as well inside of this triangle is equal to this to the side is tryingto based on the A s a assumption. So I'm remembering here, based on a s a her angle side angle Still, uh, the trying off triangles are the same. Okay, so we have a distance x, and you have the same distance x here. Um, So, um, also we did so we were able to find out the lowest point of the mirror. Other were able to find the maximum point where the ray of light hitting the lowest point in me goes to our eyes and were able to see it now in order to see her top most top heart. It's pretty much you have to. You can draw this parallel, Ray, and then have the rain reflect back. So we reflects on itself, and that's that's gonna be that's gonna be your top most part. I'm sorry. That's gonna be the part that you're able to see of yourself at the very top. So that means that ah, this this whole distance your distance of acts is actually the length of the mirror that you'll that you'll need for an object that's twice that size tonight, considers or rule here that for a mirror that sub size X a mirror that allows you to see the top of your object. Because if you if a ray of light from the top of the object came down our king from this angle, it would reflect, and you wouldn't be able to see it. If it came from a downward angle, you will reflect back and you wouldn't be able to see it. Um, but the ray of light that allows you to see yourself in the mirror is the one that's parallel to the mirror. If your eyes are up here, of course, and you're looking at your top this part. So the mirror Onley has to be between this top most part and half of your object. So let me write this again. We're only has to be 1/2 of your object length for object tight. Okay, so we're able to, um, pretty much solved the question, which is? What's the minimum mirror high for a person that's 100 ah, 78 centimeters tall, and the answer is half of that length or 89 centimeters needed for your mirror A minimum or this is also saying that the minimum mirror length that you need forgiven object length is 1/2. So you need at least 1/2 the length of your object length in order for you to see um, in order for you to see your complete object are complete image in the mirror. So that's the answer to this question. You only half of your height, how half of your object type to see your total image in the mirror.