A Beautiful Square and Circle Puzzle | Finding the Radius |

[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, a circle of radius R intersects a square of side length 24.

[00:00:16]The circle passes through the midpoint of the top side of the square.

[00:00:21]And the circle is tangent to the bottom side of the square at its bottom right vertex.

[00:00:27]Our goal is to find the radius of the circle.

[00:00:30]Before we proceed, please take a moment to hit the like button and subscribe to my channel.

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[00:00:49]To solve this problem, first, let the vertices of the square be A, B, C, and D.

[00:01:00]The circle intersects the top edge of the square at its midpoint.

[00:01:05]Let's call this point M.

[00:01:07]The circle is tangent to the bottom edge of the square at the bottom right vertex.

[00:01:12]Because the bottom edge of the square is a horizontal line, the radius drawn to this point of tangency must be perfectly vertical.

[00:01:21]Therefore, the center O lies on a vertical line passing through the right edge of the square.

[00:01:28]Because point O is the center of the circle and point B is the bottom right corner where the circle touches.

[00:01:35]Therefore, the vertical segment from O to B is exactly equal to the radius.

[00:01:42]Since the entire right side from B to C is 24, the remaining upper portion from O to C is simply the total length minus the radius, which is 24 minus R.

[00:01:58]M is the midpoint of the top edge DC.

[00:02:02]Since the total width is 24, the segment MC is exactly half of that.

[00:02:08]Which is equal to 12.

[00:02:11]Next, let's connect O and M with a straight line.

[00:02:15]Because point M is a point where the circle intersects the top edge, the straight line segment connecting the center O to point M is also a radius of the circle, which is simply R.

[00:02:27]Now, let's focus on triangle OCM in the top right corner.

[00:02:33]Because each interior angle of a square is 90°.

[00:02:38]Triangle OCM is a right triangle.

[00:02:41]Using the Pythagorean theorem, OM² is equal to OC².

[00:02:50]+ MC².

[00:02:55]OM is equal to R.

[00:03:00]OC is equal to 24 - R.

[00:03:08]And MC is 12.

[00:03:12]Substituting these values in the above expression will give us R² is equal to 24 - R².

[00:03:28]+ 12².

[00:03:30]Simplifying this will give us R² is equal to 576 - 48R + R².

[00:03:47]+ 144.

[00:03:51]Rearranging this will give us the square of r minus the square of r plus 48 r is equal to 576 plus 144.

[00:04:11]The square of r minus the square of r is zero.

[00:04:16]576 plus 144 is equal to 720.

[00:04:22]And we are left with 48 r is equal to 720.

[00:04:31]Dividing through by 48 48 will cancel out 48 and 720 divided by 48 is equal to 15.

[00:04:47]And we are left with r is equal to 15.

[00:04:52]Hence, the radius of the circle is 15 units.

[00:04:57]Thanks for watching.

[00:04:58]Don't forget to like and subscribe for more videos.