Can YOU Find the Radius of This Circle?

Hello my lovelies, it's Susanna and today I want to show you how to solve this problem where we have to find the radius of this circle. Our circle lies inside here, it touches the side here, this side here, this side here, and this here as well. We have a right angle here, a right angle here, and the length of this side is given by three, and the length of this side is given by seven. Okay, this is all the information we have, so let's see if we can find our radius. Okay, where do we start? Usually when we have these touching points, they are very important to solve problems like this. So I would focus on the radius first because we can draw it always from the center to the edge of the circle. So we can draw it for example here from the center to the edge of the circle down here to this touching point. This is our radius, and now we also know something about this angle here. Because this is a tangent to our circle at this point here, then the radius is always perpendicular to this tangent at this touching point. So we know that we have a right angle right here. Let's do this with the other touching points as well. So I want to draw my radius here to the side, and this time this is my tangent, so I know that I have a right angle here. Then we have this other touching point up there, so we connect it. This is my radius again, this time this is my tangent, and I have a right angle here. And the last touching point here, I also connect this, so this is my radius again, and this time this is my tangent, and I have a right angle here. Okay, many right angles, our radius is everywhere here. What do we know? Because we have right angles all over here, we know the length of this part here, because this is just my radius, so I know that this here is of length R. Then I can also say how long this part here is, because I know that the entire side, so this whole thing, is of length 7. But I'm not interested in the entire side, I only want this part here, so I can just subtract this part here, so my R, to get the length of this. So the length of this is 7, the entire side, minus this part here, so minus R. I can do the same thing up here, because this is my radius again, because all of these are right angles here, so I know that this here is of length R. Then I can also say how long this small part here is, because the entire side is of length 3, so I don't want the entire side, I have to subtract this part here then, so my R, to get the length of just this part here. What about this length? Well, this is from up here, my radius, plus my radius again, so this is 2R. Okay, so now we know some of the lengths. What about the length of this side here? It looks like it could be part of this triangle here, which would be a right triangle. We can create this triangle if we just draw a line from up there, down here, nice and perpendicular, so that we have a right angle here, and now we created this right triangle, and we could work with the Pythagorean theorem in here, which would be great, but for that we would need the sides of this triangle. We've already figured out the length of this side here, which is 2R, but what about the length of this side here? Well, we know that the entire side is of length 7, but we are not interested in this part here, but this has the same length as this part up here, because we drew this line perpendicular down here. If we have length 3 here, and we have our right angles here everywhere, we have length 3 down here again, so if the entire side has length 7, this here is 3, then we know that our green part has length 4. Okay, this is everything we know so far, and we still need the length of this side of our triangle. Maybe we start with this length here, so from this touching point down here to this corner. This here, this line, is a tangent to our circle, so this line touches my circle at this point, but going out from this corner here, I could also take this line here up to this touching point. This line here is also a tangent to my circle at this point here, so coming out of the same point here, I have a tangent to my circle here, and another tangent to my circle here, and the thing is that these lines have always the same length if they come out of the same point, so how long is this green part here from the corner to the touching point? This is my 7 minus r, so I immediately know that my other tangent from the corner to this touching point is also of length 7 minus r. You can prove that by using similar triangles, so if you wanted to prove that, you can create a triangle from this corner to this center here, then you have a triangle here, and another triangle here, and you just show that these triangles are similar to each other, and then this side here has to have the same length as this side here. Okay, so now we know the length of this part here. Do we also know the length of this tiny part up here? Yes, we do, with the same explanation. This time, it is from this corner here to this touching point, so my orange line here is also a tangent to my circle, but starting from this point here, and if I do the same to the left, so starting at this point and going to this touching point, this is also a tangent starting at this point. I know the length of this tangent, it's 3 minus r, and because of the same explanation as we had here, we know that this length has to be the same as this one here, so also 3 minus r. So, I want to have the length of the entire side, so I just add these two, so 3 minus r plus 7 minus r, 3 plus 7 equals 10, and negative r minus another r equals negative 2r, so in total, the entire side of my triangle has length 10 minus 2r. So, let's take our right triangle now. We have length 2r, length 4, and length 10 minus 2r, so I drew the triangle here now again with all these lengths, and now let's apply the Pythagorean theorem to our right triangle. For this, we first have to find the hypotenuse, that is always across my right angle, this is my hypotenuse, and the Pythagorean theorem then says take one of the sides, so in our case the 2r, and square this thing, plus take the other side and square it, so 4 squared, and this is equal to the hypotenuse squared, so we take the 10 minus 2r and square this whole thing, and now we only have to solve this equation only. It's going to be some steps, but that's fine, that's what we're all here for. Let's get rid of the parentheses here, we can just square the 2 and square the r, so 2 squared equals 4, and r squared equals r squared, plus 4 squared equals 16, and on the other side here, we have to be careful if we want to square these parentheses, because of this minus here in between, we are not allowed to just take this number and square it, and take this number and square it, so instead, to square these parentheses, we could write it, instead of the square, we can just multiply these parentheses by the same parentheses again, by multiplying each element of the first parentheses by each element of the second. So let's multiply. 10 times 10 equals 100. 10 times negative 2r equals negative 20r. Then the next element, negative 2r times 10 equals negative 20r, and the last negative 2r times negative 2r. Negative times negative equals positive. 2r times 2r equals 4r squared. Okay, maybe on the right side, we can simplify these here, because we have negative 20 of them, minus 20 more of them, which equals negative 40 of our r's, so I'm going to write negative 40r here, and take my r squared part a little closer. Let's get rid of the r squared part, so let's subtract 4r squared on both sides of the equation. What is left then? This cancels out, and only the 16 is left. Here we had the 100, we subtract the 40r, and this cancels out as well, which is great. No quadratic part here. We want to solve for r, so maybe let's subtract the 100 then on both sides, so that we get 16 minus 100 equals negative 84, and on the other side, this cancels out, and only the negative 40r are left. To solve for r, we only have to get rid of the negative 40, so let's divide both sides by negative 40, here and here. On the right side, this will cancel out, and only my r will be left. On the other side, this is my fraction. Let's make it a little nicer. The minus cancels out, and 84 over 40, we can reduce this fraction, because both numbers are divisible by 4. 84 over 4 equals 21, and 40 over 4 equals 10, so I have in total 21 over 10 for my radius. We were looking for the radius. We can write it as a decimal number as well for the radius, so 21 over 10 equals 2.1, and this is the result for our problem. I'm curious how you solve this problem, so please let me know in the comments. I wish you a wonderful day, and I hope to see you in one of my next videos. Take care!