I Liked This Brilliant Catriona Agg Puzzle!

Added: 2026-05-06

[00:00:00]Hey guys, this looks like a fun one.

[00:00:01]We're given a quarter circle with a radius of two. Another quarter circle with a radius of three. A semicircle with a radius of 1. These two are tangent at this point. These two are tangent at this point. And these two are tangent at this point. And it's asking us to find this angle made up of these three tangent points. If you want to try it on your own, pause it right now because I'm going to solve it in three, two, one. First, we can label this angle A and this angle B°. And we can make our first equation. The three of these will make 180°. So, we can say A plus question mark + B is equal to 180. This looks important. Let's put a box around it. And let's move it down here. Next, let's look at this green quarter circle.

[00:00:43]Since this radius is equal to two, this radius would also equal two. And if we connect this point to this point, that'll also be a radius of the quarter circle. it's also going to be equal to two. Next, we can focus on this triangle. It is an isosesles triangle.

[00:00:59]These two sides are both equal to two.

[00:01:01]And then in every single isoclesles triangle, the two angles opposite the congruent sides are congruent. Since the angle opposite this side of two is a, the angle opposite this side of two will also be a. Now, let's call the last angle c°. And now we have another equation. The sum of the interior angles of a triangle is 180°. So we know that 2 a + c is 180. And now we can do the same thing on this side. Since this is three, this will also be three. And then if we draw this from here to here, it'll also be three. And then we can focus on this triangle. It's also an isoclesles triangle with two side lengths of three.

[00:01:38]This angle is opposite this side of three and it's b°. So this angle opposite this side of three will also be b°. And then the last angle, let's label it d. And we have another equation. 2B + D is equal to 180. Next, let's extend this all the way up here and this all the way up here. And they're going to intersect at the center of this semicircle. We can clean it up a little bit and put a dot there. Here are the notes explaining why we know this hits at the center. Anytime you have two circles or semicircles or quarter circles, if you connect the two centers, it always goes through the tangent point. or if you go through the center of one through the tangent of another, it'll always hit the center of the other one. And that's going to be the same for this and for this. So they're going to intersect at the center of this blue or purple semicircle. Since this semicircle has a radius of 1, this radius is equal to 1. And this radius is equal to 1.

[00:02:33]Next, we can focus on this triangle.

[00:02:36]This side length is equal to 1 + 2, which is equal to 3. This side length is 1 + 3, which is equal to 4. And this side is 2 + 3 which is equal to 5. Now this is a 345 triangle. Any 345 right triangle satisfies the pyagorean theorem. 32 + 42= 5^2. So this is a right triangle and the angle opposite the 5 is the right angle. Now once again this triangle has 180° in it. So we can say C + D + 90= 180. And now what are we going to do next? We have four variables and three equations. I got to try to figure out how we can solve this. Okay, I figured it out. Let's move this down here. And we're going to get C plus D alone by subtracting 90 from both sides.

[00:03:20]On the left hand side, the positive 90 and negative 90 are going to cancel each other out. And then on the right hand side, 180 minus 90 is equal to 90. Let's smush this together and move it down here. And for these two equations, we're going to add the two rows together. On the left hand side, we're going to have 2 A + 2 B + C + D. And on the right hand side, 180 plus 180 is equal to 360.

[00:03:43]Next, we can move this stuff down here.

[00:03:45]And then in the place of this C plus D, we can plug in 90. And we can copy down everything else. Now, each of our terms is divisible by two. So, let's divide everything by two. After we divide both sides by two, on the left hand side, we'll have a + b + 45. And on the right hand side, 360 / two is equal to 180.

[00:04:04]Let's smush everything together and subtract 45 from both sides. On the left hand side, we're going to be left with a + b. And on the right hand side, 180 minus 45 is 135. And a plus b is what was missing in our equation. Let's bring everything up here, break this out of the box, and rearrange it like this. In the place of the a + b, we can plug in 135. To get the question mark all by itself, we can subtract 135 from both sides. And that'll leave us with question mark is equal to 180 minus 135, which is 45. And that is the answer to our question. Let's give it a label of degrees and put a box around it. In this given diagram, the angle made up of these tangent points is equal to 45°.

[00:04:47]How exciting. I've done so many of these Katrina a puzzles. It was really fun to see one I haven't done yet. It was a cool one. And it's the same thing for Brilliant. I'm always excited to find a course I haven't done yet. I saw this course called Probability and Chance, and I love probability and chance. I was surprised I hadn't gone through it yet.

[00:05:03]I already know pretty much all the material they do on probability and chance, but I really love seeing how they go about explaining it. They have world-class teachers from MIT, Harvard, and Stanford. You can tell they put a lot of thought in how they set up the lessons, how they use the interactive elements, the helpful hints, and the suggestions along the way. And no matter what topic you're interested in, whether it's math, science, data analysis, computer science, or programming, Brilliant has you covered. If you'd like to try Brilliant for free for a full 30 days, visit brilliant.org/andandymath.

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