Can You Find Angle X?

[00:00:00]What's up everyone?

[00:00:01]Welcome back.

[00:00:03]Today we've got a beautiful geometry puzzle.

[00:00:07]It looks innocent at first, but the solution hides a clever circle theorem.

[00:00:13]Let's see if we can crack it together.

[00:00:16]Here's the setup.

[00:00:18]We have a rectangle ABCD.

[00:00:21]Along the bottom side sits a semicircle whose diameter is AM.

[00:00:28]The arc of the semicircle just touches the top side of the rectangle at point P.

[00:00:34]It also meets the right side of the rectangle at point Q.

[00:00:39]Now, here's the key piece of information.

[00:00:42]Point Q divides the right side exactly into two equal parts.

[00:00:47]So, BQ is equal to CQ. Our goal is to find the measure of angle BAQ.

[00:00:56]Let's call that angle X.

[00:00:59]The first step is to locate the center of the semicircle. Call the center O.

[00:01:06]Now, draw the radius OP.

[00:01:09]Since the top side of the rectangle is tangent to the semicircle at P, the radius OP is perpendicular to that side.

[00:01:18]That means the height of the rectangle is exactly the radius of the semicircle.

[00:01:23]Let's call the radius R.

[00:01:26]So, BC equals R.

[00:01:32]But Q is the midpoint of BC.

[00:01:36]Therefore, BQ equals R divided by 2.

[00:01:42]Now join O to Q.

[00:01:44]Since Q lies on the semicircle, OQ is another radius.

[00:01:50]So, OQ also has length R.

[00:01:55]Look at right triangle OBQ.

[00:01:58]Let's call the angle at O alpha.

[00:02:03]Using the definition of sine, sine of alpha equals the opposite side divided by the hypotenuse.

[00:02:12]The opposite side is BQ, which is R divided by 2.

[00:02:17]The hypotenuse is OQ, which is R.

[00:02:22]So, sine of alpha equals the quantity R divided by 2 divided by R, which simplifies to 1/2. And when the sine of an angle is 1/2, the angle is 30°.

[00:02:41]So, alpha equals 30°.

[00:02:46]Now comes the beautiful finishing step.

[00:02:49]We'll use the inscribed angle theorem.

[00:02:53]The central angle MOQ measures 30°. So, the arc MQ also measures 30°.

[00:03:02]An inscribed angle standing on that same arc is exactly half the measure of the central angle.

[00:03:10]The angle standing on arc MQ is angle BAQ.

[00:03:15]Therefore, angle BAQ equals half of 30°, which is 15°.

[00:03:25]And that's our final answer.

[00:03:28]If you enjoyed this problem, don't forget to like the video, subscribe for more geometry challenges, and I'll see you in the next one.