The Semicircle Geometry Puzzle That Confuses Adults!|

[00:00:00]Hello everyone and welcome back to my channel.

[00:00:03]In today's video, we are going to solve another interesting geometric problem.

[00:00:08]In this question, we are given two nested semicircles sharing a common baseline.

[00:00:15]The smaller semicircle has a diameter BC.

[00:00:20]The larger semicircle has a diameter AD, where A, B, C, and D are collinear.

[00:00:28]The line segment AF starts at A, is tangent to the smaller semicircle at point E, and intersects the larger semicircle at point F.

[00:00:39]Given that AB is equal to 4 units, CD is equal to 9 units, AE is equal to 8 units, and EF is equal to X.

[00:00:53]Our goal is to determine the length of X.

[00:00:56]Before we proceed, please take a moment to hit the like button and subscribe to my channel.

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[00:01:08]Also, feel free to pause the video and give it a try, and tell us your answer in the comment section.

[00:01:15]To solve this problem, first, let's find the diameter of the smaller semicircle.

[00:01:21]To do this, we will use the tangent-secant theorem to the smaller circle.

[00:01:28]The theorem states that if a tangent segment and a secant line are drawn to a circle from an exterior point, the square of the tangent segment is equal to the product of the external secant segment and the entire secant segment.

[00:01:43]For the point A outside the smaller circle, the tangent segment AE is equal to 8.

[00:01:51]The secant line lies along the diameter line AD, making the external segment AB equal to 4.

[00:01:59]And the entire secant segment AC.

[00:02:04]Applying the theorem AE squared is equal to AB times AC.

[00:02:16]Substituting each value in the equation will give us 8 squared is equal to 4 times AC.

[00:02:27]Simplifying this will give us 64 is equal to 4 AC.

[00:02:38]Dividing through by 4 4 will cancel out 4 and 64 divided by 4 is 16.

[00:02:50]And we are left with AC is equal to 16.

[00:02:57]Since AC is equal to AB plus BC it follows that 16 is equal to 4 plus BC.

[00:03:15]Rearranging this will give us BC is equal to 16 minus 4.

[00:03:24]16 minus 4 is 12.

[00:03:28]Therefore BC is equal to 12.

[00:03:34]This tells us that the diameter of the smaller circle, segment BC, is exactly 12.

[00:03:41]And its radius is half of that, which is 6.

[00:03:45]Now let O be the center of the smaller semicircle.

[00:03:51]Let's construct two key lines.

[00:03:54]First, if we draw a line from the center O to the point of tangency E, it meets the line AF at a perfect 90° angle.

[00:04:03]This creates a small right-angle triangle AEO.

[00:04:08]Next, let's connect D to F.

[00:04:12]Since AD is the diameter of the larger semicircle, the inscribed angle angle AFD is equal to 90°.

[00:04:21]by Thales' theorem.

[00:04:23]This gives us a larger right-angle triangle AFD.

[00:04:28]Now, comparing the small right triangle AEO and the large right triangle AFD, both triangles perfectly overlap at the far left corner, meaning they share the exact same angle at point A.

[00:04:44]Both triangles contain a 90° right angle.

[00:04:49]According to the angle-angle similarity rule of geometry, when two triangles share two identical angles, they are similar triangles.

[00:04:57]Therefore, triangles AEO and AFD are similar.

[00:05:10]Because the triangles are similar, the ratios of their corresponding sides must be equal.

[00:05:16]This follows that AE over AO is equal to AF over AD.

[00:05:27]AE is equal to 8.

[00:05:33]AO is equal to AB plus BO, which is equal to 4 + 6.

[00:05:46]And 4 + 6 is equal to 10.

[00:05:51]AF is equal to AE plus EF which is equal to 8 + X.

[00:06:05]AD is equal to AB plus BC plus CD which is equal to 4 + 12.

[00:06:21]plus 9 And 4 + 12 + 9 is equal to 25.

[00:06:29]Substituting these values into our similarity ratio will give us 8 over 10 is equal to 8 + X over 25.

[00:06:44]By cross multiplication, we have 10 * 8 + X is equal to 8 * 25.

[00:06:57]Simplifying this will give us 80 plus 10X is equal to 200.

[00:07:13]And rearranging this will give us 10X is equal to 200 minus 80.

[00:07:24]200 minus 80 is equal to 120.

[00:07:31]So, we are left with 10X is equal to 120.

[00:07:37]Dividing through by 10 10 will cancel out 10.

[00:07:45]And 120 divided by 10 is 12.

[00:07:50]Hence, X is equal to 12 units.

[00:07:54]Thanks for watching.

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