Okay, so I need to find the radius of the smaller semicircle in this geometry problem. Let me try to visualize the setup first. There's a circle with center Q, and two radii AQ and BQ form a right angle. That means AQ and BQ are perpendicular to each other, right? So, points A and B are on the circumference of the main circle, and the angle between them at Q is 90 degrees. The main circle has a radius of 14 inches. 

Now, there are two smaller regions that are tangent semicircles. Hmm, tangent semicircles... I need to figure out where these semicircles are located. Since AQ and BQ form a right angle, the semicircles must be drawn inside the main circle. Maybe they are touching each other and also tangent to the main circle? Let me sketch this mentally. If AQ and BQ are radii forming a right angle, then the triangle AQB is a right-angled triangle with legs of length 14 each. Wait, no, AQ and BQ are radii, so they are both 14 inches long. But the angle between them is 90 degrees, so the length of AB would be 14√2 by the Pythagorean theorem. But maybe that's not directly relevant here.

The two smaller regions are tangent semicircles. Since they are tangent, they must touch each other at exactly one point. Also, they are inside the main circle. Let me think. If they are semicircles, each must have a diameter lying along the radii AQ and BQ. Wait, but if they are tangent, maybe their centers are somewhere along AQ and BQ. Let me consider that.

Suppose the semicircle on AQ has its diameter along AQ. Then its center would be the midpoint of AQ, which is at 7 inches from Q. Similarly, the semicircle on BQ would have its center at the midpoint of BQ, which is also 7 inches from Q. But if these two semicircles are tangent to each other, the distance between their centers should be equal to the sum of their radii. Wait, but if both semicircles are drawn with radius 7, then the distance between their centers is the distance between the midpoints of AQ and BQ. Since AQ and BQ are perpendicular, the distance between the midpoints would be sqrt((7)^2 + (7)^2) = 7√2. But the sum of their radii would be 7 + 7 = 14. But 7√2 is approximately 9.899, which is less than 14. So that can't be. Therefore, maybe the semicircles are not drawn with diameters along AQ and BQ, but perhaps with diameters along other lines.

Wait, maybe the semicircles are drawn such that their diameters are along the sides of the right angle, but not necessarily the entire radius. Let me think again. If the semicircles are tangent to each other and also tangent to the main circle, then their centers must lie along the angle bisector of the right angle at Q. Wait, the angle at Q is 90 degrees, so the angle bisector would be a line at 45 degrees from each radius. So maybe the centers of the semicircles lie along this bisector. Hmm, that could be a possibility.

Let me denote the radius of the smaller semicircle as r. Then, the center of this semicircle would be somewhere along the bisector, at a distance of (14 - r) from Q, since the semicircle is tangent to the main circle. Because the main circle has radius 14, and the semicircle is inside it, so the distance from Q to the center of the semicircle is 14 - r. But the center of the semicircle is also along the bisector. Let me confirm that. If the semicircle is tangent to the main circle, then the distance from Q to the center of the semicircle is 14 - r. Also, since the semicircle is tangent to the other semicircle, the distance between their centers must be equal to the sum of their radii. But if there are two semicircles, maybe both with radius r? Wait, the problem says "two smaller regions are tangent semicircles". So maybe both are semicircles with the same radius r. Then, the distance between their centers would be 2r. But the centers are located along the angle bisector, each at a distance of 14 - r from Q. Wait, but if both centers are on the bisector, then the distance between them would be 2*(14 - r)*sin(45°), since the angle between them is 45 degrees. Wait, no, actually, since the bisector splits the right angle into two 45-degree angles, the centers are on the bisector, so the angle between the lines from Q to each center is 45 degrees. Therefore, the distance between the two centers can be found using the law of cosines: sqrt((14 - r)^2 + (14 - r)^2 - 2*(14 - r)*(14 - r)*cos(45°)). 

But since the semicircles are tangent to each other, the distance between their centers should be equal to the sum of their radii, which is r + r = 2r. So we can set up the equation:

sqrt(2*(14 - r)^2 - 2*(14 - r)^2*(sqrt(2)/2)) = 2r

Simplify the expression inside the square root:

First, factor out 2*(14 - r)^2:

sqrt[2*(14 - r)^2*(1 - sqrt(2)/2)] = 2r

Take out the 2*(14 - r)^2 from the square root:

sqrt(2)*(14 - r)*sqrt(1 - sqrt(2)/2) = 2r

Hmm, this seems complicated. Let me check if this approach is correct. Alternatively, maybe the centers are not both along the bisector. Let me reconsider.

Wait, maybe the two semicircles are each tangent to one radius and the main circle. For example, one semicircle is tangent to radius AQ and the main circle, and the other is tangent to radius BQ and the main circle. Then, the centers of these semicircles would lie along AQ and BQ respectively. Let me think. If the semicircle is tangent to AQ and the main circle, then its radius would be something else. Wait, but the problem says the two smaller regions are tangent semicircles. So maybe each semicircle is tangent to one radius and the main circle.

Let me try this approach. Suppose there is a semicircle tangent to AQ and the main circle. The center of this semicircle would be somewhere inside the main circle, along the radius AQ. Let the radius of this semicircle be r. Then, the distance from Q to the center of the semicircle would be 14 - r, because the semicircle is tangent to the main circle. Also, since the semicircle is tangent to AQ, the distance from the center of the semicircle to AQ must be equal to its radius r. But if the center is along AQ, then the distance from the center to AQ is zero. Wait, that can't be. Hmm, maybe I got that wrong.

Wait, if the semicircle is tangent to AQ, then the center must be at a distance r from AQ. But if the center is on AQ, then the distance from the center to AQ is zero. So that can't be. Therefore, the center must be offset from AQ. Wait, but if the semicircle is tangent to AQ and lies inside the main circle, then the center of the semicircle must be at a distance r from AQ, and also at a distance 14 - r from Q. So if we have a semicircle tangent to AQ, its center is at (r, 0) in a coordinate system where Q is at (0,0), AQ is along the x-axis, and BQ is along the y-axis. Then, the distance from the center (r, 0) to Q is sqrt(r^2 + 0^2) = r. But the semicircle is tangent to the main circle, which has radius 14. So the distance from Q to the center of the semicircle must be 14 - r. Therefore, r = 14 - r, which gives 2r = 14, so r = 7. But that's the radius of the semicircle, but the problem states that the two smaller regions are tangent semicircles. If each has radius 7, then they are not smaller. So that can't be right.

Wait, maybe the semicircles are not tangent to AQ and BQ, but to each other and the main circle. Let me try that. Suppose there is a semicircle in the sector AQB, tangent to AQ, BQ, and the main circle. Then, the center of this semicircle would be somewhere inside the sector. Let me model this. Let's set up a coordinate system with Q at (0,0), A at (14,0), and B at (0,14). The semicircle is tangent to the x-axis, y-axis, and the circle of radius 14 centered at Q. Let the center of the semicircle be at (r, r), since it's tangent to both axes. Then, the distance from the center (r, r) to the origin Q is sqrt(r^2 + r^2) = r√2. This distance must be equal to 14 - r, because the semicircle is tangent to the main circle. Therefore:

r√2 = 14 - r

Solving for r:

r√2 + r = 14

r(√2 + 1) = 14

r = 14 / (√2 + 1)

Multiply numerator and denominator by (√2 - 1):

r = 14(√2 - 1) / [(√2 + 1)(√2 - 1)] = 14(√2 - 1) / (2 - 1) = 14(√2 - 1)

But this is approximately 14*(1.414 - 1) = 14*0.414 ≈ 5.796 inches. However, the problem mentions two smaller semicircles. If there's only one such semicircle, perhaps there are two of them, each in the two smaller sectors. Wait, but the sector between AQ and BQ is a quarter-circle, so if there is one semicircle in that sector, maybe there is another one in the opposite sector? Wait, the angle is 90 degrees, so the two smaller regions are in the two quadrants formed by AQ and BQ. Wait, but AQ and BQ are radii forming a right angle, so the regions outside the semicircles would be the two smaller regions. Hmm, maybe the two semicircles are each tangent to the main circle and one of the radii. Wait, but the problem says "the two smaller regions are tangent semicircles". So perhaps each semicircle is tangent to one radius and the main circle. Then, each semicircle would have its center at a distance of 14 - r from Q, and the distance from the center to the respective radius must be equal to r. For example, for the semicircle tangent to AQ, its center is at (14 - r, 0), and the distance from this center to AQ (the x-axis) is 0, which can't be. Wait, that's not possible.

Alternatively, maybe the semicircles are not aligned with the radii. Let me think again. Suppose the semicircles are located in the two smaller regions formed by the radii AQ and BQ. Each semicircle is tangent to the other semicircle and tangent to the main circle. So each semicircle is in one of the two smaller quadrants. Let's focus on one of them. Let's say the semicircle is in the quadrant between AQ and BQ. It is tangent to AQ, BQ, and the main circle. Then, its center would be at some point (r, r), as before. The distance from the center to Q is sqrt(r^2 + r^2) = r√2. This distance must equal 14 - r, since the semicircle is tangent to the main circle. So:

r√2 = 14 - r

Which gives r = 14 / (√2 + 1) = 14(√2 - 1)/1 = 14√2 - 14 ≈ 14*0.414 ≈ 5.796 inches. But the problem mentions two smaller semicircles, so perhaps there are two of these, each in their respective quadrants. But then, the radius would be the same for both. Wait, but the problem says "the two smaller regions are tangent semicircles", so maybe each has the same radius. But then the answer would be 14√2 - 14, which is approximately 5.796, but expressed as a common fraction. Wait, 14(√2 - 1) is already simplified, but maybe we need to rationalize it differently. Wait, 14(√2 - 1) is the exact value. But the problem says "Express your answer as a common fraction." So maybe this approach is incorrect. Alternatively, perhaps the two semicircles are not both tangent to the main circle. Wait, the problem states "the two smaller regions are tangent semicircles". So maybe each semicircle is tangent to the other semicircle and to one of the radii. Let me consider that. Let's say there's a semicircle in the sector AQB, tangent to AQ, and another semicircle in the sector BQA (but that's the same sector), so maybe not. Alternatively, maybe the two semicircles are in the two smaller regions formed by the radii AQ and BQ, each tangent to the other semicircle and to the main circle. Let me try to model this.

Let me consider the two semicircles as lying in the two quadrants formed by AQ and BQ. Each semicircle is tangent to the main circle and to the other semicircle. Let's focus on one semicircle first. Suppose the semicircle is in the first quadrant (between AQ and BQ). Let its radius be r. The center of this semicircle must be at a distance of 14 - r from Q, since it's tangent to the main circle. Also, the semicircle is tangent to the x-axis (AQ) and y-axis (BQ). Therefore, the center of the semicircle is at (14 - r, 14 - r). Wait, but the semicircle is only in the first quadrant, so if it's a semicircle, it's either above the x-axis or to the right of the y-axis. Wait, but if it's tangent to both axes, then the center would be at (r, r), but that's if the semicircle is in the corner. Wait, no. If the semicircle is tangent to both axes, then its center would be at (r, r), but since it's a semicircle, the diameter would be along one of the axes. Wait, maybe not. Let me think. If the semicircle is tangent to both the x-axis and y-axis, then its center must be at (r, r), and the semicircle would have to be in the first quadrant. But if it's a semicircle, then it's either the upper half or the right half. Wait, but the problem says "two smaller regions are tangent semicircles". So maybe each semicircle is a quarter-circle? No, the problem says semicircles, so half-circles. Therefore, the semicircles must be in such a way that they are tangent to both the x-axis and y-axis, and also tangent to the main circle. Wait, but if a semicircle is tangent to both axes, then its center must be at (r, r), and the semicircle would have a diameter either along the x-axis or the y-axis. Wait, but if the semicircle is tangent to the x-axis, then its center is at (a, r), where a is the x-coordinate. Similarly, if it's tangent to the y-axis, the center is at (r, b). But if it's tangent to both, then the center is at (r, r). But then, the semicircle would have to be either the upper half or the right half. Let's say it's the upper half. Then, the semicircle would be the set of points (x, y) where x = r, y ≥ r, and the distance from (r, r) to any point on the semicircle is equal to the radius. Wait, no. If the center is at (r, r) and the semicircle is the upper half, then the semicircle would be the set of points (x, y) where (x - r)^2 + (y - r)^2 = r^2, with y ≥ r. But then, the semicircle is tangent to the x-axis at (r, 0), but in this case, the center is at (r, r), so the distance from the center to the x-axis is r, which is equal to the radius. So yes, it's tangent to the x-axis. Similarly, it's tangent to the y-axis at (0, r). Wait, but the semicircle is only the upper half. So it's tangent to the x-axis at (r, 0) and the y-axis at (0, r). But then, the semicircle is in the first quadrant, and the entire semicircle is in the first quadrant. But if the center is at (r, r), then the distance from the center to the origin is sqrt(r^2 + r^2) = r√2. Since the semicircle is tangent to the main circle of radius 14, the distance from the center to the origin must be equal to 14 - r. Therefore:

r√2 = 14 - r

Which is the same equation as before. Solving:

r√2 + r = 14

r(√2 + 1) = 14

r = 14 / (√2 + 1) = 14(√2 - 1) / ( (√2 + 1)(√2 - 1) ) = 14(√2 - 1)/1 = 14√2 - 14

But this is approximately 5.796, which is less than 7, so it's a smaller semicircle. But the problem mentions two smaller regions. So if there are two such semicircles, each in the two smaller quadrants (but there's only one quadrant in the right angle), wait, no. The semicircle we just found is in the quadrant between AQ and BQ. If there's another semicircle, maybe in the opposite quadrant, but the opposite quadrant would be the other side of the main circle. But the problem says "the two smaller regions are tangent semicircles", so probably there are two such semicircles, each in their own quadrant. Wait, but in the main circle with center Q, there are four quadrants. The two semicircles are in the two quadrants adjacent to the right angle, i.e., the first and third quadrants? Wait, no. If AQ and BQ are radii forming a right angle, say AQ is along the positive x-axis and BQ is along the positive y-axis, then the two smaller regions would be the two quadrants between the main circle and the radii. Wait, perhaps the two semicircles are in the two quadrants that are adjacent to the right angle between AQ and BQ. Wait, but the main circle is divided into four quadrants by AQ and BQ, and the two semicircles are in two of those quadrants. But the problem states "the two smaller regions are tangent semicircles". So maybe the two semicircles are each in their own quadrant. But each semicircle would have the same radius, as per the symmetry. Therefore, the radius would be 14(√2 - 1). But let me check if this is correct.

Wait, if the radius is 14(√2 - 1), then the center is at (r, r). The distance from the center to the origin is r√2 = 14(√2 - 1) * √2 = 14*(2 - √2). But 14*(2 - √2) is approximately 14*(2 - 1.414) ≈ 14*0.586 ≈ 8.204. But the radius of the main circle is 14, so 14 - r = 14 - 14(√2 - 1) = 14 - 14√2 + 14 = 28 - 14√2 ≈ 28 - 19.796 ≈ 8.204. Therefore, the distance from the center to the origin is indeed 14 - r, which matches the condition that the semicircle is tangent to the main circle. Therefore, this seems correct. So the radius of each semicircle is 14(√2 - 1). But the problem asks for the radius of the smaller semicircle. Since there are two smaller regions (the two quadrants adjacent to the right angle), each containing a semicircle, and both have the same radius, the answer would be 14(√2 - 1). However, the problem says "the radius of the smaller semicircle", implying there's only one. Wait, maybe I misinterpreted the problem. Let me read again: "the two smaller regions are tangent semicircles". So there are two regions, each a semicircle, and each is smaller. Therefore, each has radius 14(√2 - 1). But the problem asks for the radius of the smaller semicircle, so the answer is 14(√2 - 1). However, 14(√2 - 1) is an exact value, but the problem says "Express your answer as a common fraction". So perhaps I need to rationalize it differently. Wait, 14(√2 - 1) is already a simplified form, but maybe it can be written as (14√2 - 14)/1, which is still a fraction. Alternatively, maybe the answer is 14/(√2 + 1), which is equivalent to 14(√2 - 1)/1. So perhaps the answer is 14(√2 - 1). But let me check if there's another approach.

Alternatively, maybe the two semicircles are located in the two smaller regions formed by the tangents to the main circle. Wait, but the problem says "the two smaller regions are tangent semicircles". So each semicircle is tangent to the other semicircle and to the main circle. Let me model this. Suppose the two semicircles are located in the two quadrants adjacent to the right angle between AQ and BQ. Each semicircle is tangent to the main circle and to the other semicircle. Let the radius of each semicircle be r. The centers of the semicircles would be along the angle bisector of the right angle, which is the line y = x in the coordinate system where Q is at (0,0), A at (14,0), and B at (0,14). The centers of the semicircles would be at a distance of 14 - r from Q, and also the distance between the centers of the two semicircles would be 2r, since they are tangent to each other. Let me confirm. If both semicircles are in the two quadrants, their centers would be symmetric with respect to the line y = x. Let's denote the centers as (a, a) and (b, b). Since they are in adjacent quadrants, but actually, in this case, since they are in the same sector, perhaps not. Wait, no. If AQ is along the x-axis and BQ along the y-axis, then the two smaller regions would be the two quadrants adjacent to the right angle, but each semicircle would be in one of those quadrants. Wait, perhaps each semicircle is in one of the two smaller regions formed by the radii AQ and BQ. So each semicircle is in a quadrant of the main circle, but the two quadrants are the ones adjacent to the right angle. Wait, but the main circle is divided into four quadrants by the x and y axes. However, in this problem, AQ and BQ are radii forming a right angle, so they are like the x and y axes. The two smaller regions would be the two quadrants adjacent to the right angle, i.e., the first and third quadrants? Wait, no. If AQ and BQ are along the x and y axes, then the two quadrants adjacent to the right angle are the first and second quadrants. Wait, maybe I'm overcomplicating. Let me draw a mental picture. The main circle is divided by AQ (x-axis) and BQ (y-axis) into four quadrants. The two smaller regions are the two quadrants that are adjacent to the right angle at Q, which are the first and second quadrants. Wait, but in the problem, the two smaller regions are the two regions formed by the radii AQ and BQ. So the two regions would be the two smaller sectors created by AQ and BQ. Since AQ and BQ form a right angle, each sector is a quadrant. But the problem mentions "two smaller regions", so maybe each of these regions is a semicircle. Wait, no. If AQ and BQ are radii forming a right angle, then the two smaller regions would be the two sectors each with a 90-degree angle. But each sector is a quadrant of the main circle. However, the problem says "the two smaller regions are tangent semicircles". So perhaps each of these two sectors (each a quadrant) contains a semicircle. Each semicircle is tangent to the main circle and tangent to the other semicircle. Therefore, each semicircle is in a quadrant, tangent to the main circle and tangent to the other semicircle. Let me model this.

Let me focus on one quadrant. Let the semicircle be in the first quadrant, tangent to AQ (x-axis), BQ (y-axis), and the main circle. Wait, but earlier we found that the radius would be 14(√2 - 1). But the problem mentions two smaller regions, each being a semicircle. Wait, perhaps each semicircle is in a different quadrant. But in the first quadrant, the semicircle is tangent to both axes and the main circle. Similarly, in the second quadrant, another semicircle is tangent to both axes and the main circle. Wait, but the second quadrant is where x is negative and y is positive. But AQ and BQ are in the first quadrant. Wait, maybe not. Let me clarify. If AQ is along the positive x-axis and BQ is along the positive y-axis, then the two smaller regions would be the two quadrants adjacent to the right angle, which are the first and second quadrants. Wait, but the semicircles in the second quadrant would be tangent to the negative x-axis and positive y-axis. But the problem says "the two smaller regions are tangent semicircles". So maybe each semicircle is in a different quadrant, but both are tangent to the main circle and to each other. Let me consider that. Let the two semicircles be in the first and second quadrants. Each is tangent to the main circle, tangent to the x-axis or y-axis, and tangent to each other. Wait, but if the semicircles are in the first and second quadrants, their centers would be along the lines y = x and y = -x. Let me model this.

Let me denote the semicircle in the first quadrant as having its center at (r, r), and the semicircle in the second quadrant as having its center at (-s, s). Both semicircles are tangent to the main circle, so the distance from (r, r) to Q is sqrt(r^2 + r^2) = r√2 = 14 - r. Similarly, the distance from (-s, s) to Q is sqrt(s^2 + s^2) = s√2 = 14 - s. Also, the two semicircles are tangent to each other, so the distance between their centers must be equal to r + s. The distance between (r, r) and (-s, s) is sqrt{(r + s)^2 + (r - s)^2} = sqrt{(r + s)^2 + (r - s)^2} = sqrt{2(r^2 + s^2)}. Therefore, we have:

sqrt{2(r^2 + s^2)} = r + s

Squaring both sides:

2(r^2 + s^2) = (r + s)^2 = r^2 + 2rs + s^2

Simplify:

2r^2 + 2s^2 = r^2 + 2rs + s^2

r^2 + s^2 = 2rs

Which implies (r - s)^2 = 0, so r = s. Therefore, the two semicircles have the same radius. Therefore, both semicircles have radius r = s, and the above equation becomes:

sqrt{2(2r^2)} = 2r

sqrt{4r^2} = 2r

2r = 2r, which is always true. Therefore, this condition is satisfied for any r, but we also have the condition that the distance from each center to Q is 14 - r. Therefore, from the first quadrant center (r, r):

r√2 = 14 - r

Which gives r = 14 / (√2 + 1) = 14(√2 - 1). Similarly, the second quadrant center (-s, s) would also have s = r = 14(√2 - 1). Therefore, the radius of each semicircle is 14(√2 - 1). Therefore, the answer is 14(√2 - 1). But the problem asks for the radius of the smaller semicircle. Since both semicircles are the same size, the answer is 14(√2 - 1). But 14(√2 - 1) is approximately 5.796, which is smaller than 14, so it's a smaller semicircle. Therefore, the answer should be 14(√2 - 1). However, the problem says "Express your answer as a common fraction". Since √2 is irrational, this is not a common fraction. Therefore, perhaps there's a different approach.

Wait, maybe I made a mistake in assuming the semicircles are tangent to both axes. Let me check again. The problem states "the two smaller regions are tangent semicircles". If the semicircles are tangent to each other and to the main circle, but not necessarily to the radii AQ and BQ. Then, perhaps the two semicircles are each tangent to the main circle and to each other, but their centers are along the angle bisector. Let me model this. Suppose the two semicircles are located symmetrically with respect to the angle bisector of the right angle. Let their centers be at points along the bisector at a distance of 14 - r from Q, where r is their radius. The distance between their centers would be 2*(14 - r)*sin(45°), because they are along the bisector, which is at 45 degrees from each radius. Since they are tangent to each other, the distance between their centers must be 2r. Therefore:

2*(14 - r)*sin(45°) = 2r

Simplify:

(14 - r)*(√2/2) = r

Multiply both sides by 2:

(14 - r)√2 = 2r

14√2 - r√2 = 2r

14√2 = r√2 + 2r

14√2 = r(√2 + 2)

Therefore:

r = 14√2 / (√2 + 2)

Multiply numerator and denominator by (√2 - 2) to rationalize:

r = 14√2(√2 - 2) / [ (√2 + 2)(√2 - 2) ] = 14√2(√2 - 2) / (2 - 4) = 14√2(√2 - 2) / (-2) = -14√2(√2 - 2)/2 = -14(2 - 2√2)/2 = -14(1 - √2)

But radius cannot be negative, so take absolute value:

r = 14(√2 - 1)

Which is the same result as before. Therefore, the radius is 14(√2 - 1). But again, this is not a common fraction. Therefore, perhaps the problem expects the answer in a different form. However, the problem specifically says "Express your answer as a common fraction". Since √2 is irrational, the exact value is 14(√2 - 1), which is an exact expression but not a common fraction. Therefore, perhaps there's a misunderstanding in the problem statement. Alternatively, perhaps the problem refers to the two semicircles as being in the two smaller regions formed by the radii AQ and BQ, each being a semicircle. Wait, but if AQ and BQ form a right angle, the two smaller regions would be the two quadrants adjacent to the right angle. Each of these regions is a quadrant of the main circle. However, a quadrant of a circle is a quarter-circle, not a semicircle. Therefore, the problem might be referring to two semicircular regions within the main circle, each of which is tangent to the main circle and to each other, and each lying in one of the two smaller quadrants. But in that case, each semicircle would have a diameter equal to the radius of the main circle, but that seems inconsistent. Alternatively, perhaps the semicircles are constructed such that their diameters are along the radii AQ and BQ, but then they would be quarter-circles, not semicircles. Hmm, perhaps I need to re-examine the problem statement.

The problem states: "the two smaller regions are tangent semicircles". So each of these two regions is a semicircle. Therefore, each is a half-circle. The main circle has radius 14, and the two smaller semicircles are inside the main circle. Each semicircle is tangent to the other semicircle and to the main circle. Let me consider that each semicircle is in one of the two smaller regions formed by the radii AQ and BQ. Since AQ and BQ form a right angle, the two smaller regions are the two quadrants between AQ and BQ and the other side. Wait, but in the main circle, the two smaller regions would be the two quadrants adjacent to the right angle. Each of these quadrants is a 90-degree sector. Each semicircle would be inscribed in one of these quadrants. Therefore, the semicircle in the first quadrant would have its diameter along the x-axis from (0,0) to (r,0), and the semicircle would extend upwards. Similarly, the semicircle in the second quadrant would have its diameter along the y-axis from (0,0) to (0,r), extending to the left. But wait, if the semicircle is tangent to the main circle, then the distance from the center to Q must be equal to 14 - r. Let me check. For the semicircle in the first quadrant with diameter along the x-axis from (0,0) to (r,0), the center would be at (r/2, 0), and the radius would be r/2. Wait, but if the semicircle is above the x-axis, then the center is at (r/2, 0), and the radius is r/2. The distance from the center to Q is sqrt( (r/2)^2 + 0^2 ) = r/2. Since the semicircle is tangent to the main circle, the distance from the center to Q must be equal to 14 - r/2. Therefore:

r/2 = 14 - r/2

Which gives:

r/2 + r/2 = 14

r = 14

But this can't be, as the semicircle would coincide with the main circle. Therefore, this approach is incorrect.

Alternatively, perhaps the semicircles are not constructed with their diameter along the radii, but rather as half-circles inside the quadrants. Let me consider that the semicircle is tangent to the x-axis, y-axis, and the main circle. Then, the center would be at (r, r), and the distance from the center to Q is sqrt(r^2 + r^2) = r√2. Since the semicircle is tangent to the main circle, the distance from the center to Q must be 14 - r. Therefore:

r√2 = 14 - r

Which again gives r = 14/(√2 + 1) = 14(√2 - 1). Therefore, this seems to be the correct radius. However, since the problem mentions "two smaller regions", each of which is a semicircle, and each has radius 14(√2 - 1). Therefore, the answer is 14(√2 - 1). But the problem says "Express your answer as a common fraction", which suggests a fractional answer. However, since √2 is irrational, the exact value is 14(√2 - 1). Therefore, perhaps the problem expects the answer in terms of √2, even though it's not a common fraction. Alternatively, maybe there's a miscalculation.

Wait, let me check again. If the semicircle is tangent to the x-axis, y-axis, and the main circle, then the center is at (r, r), distance from center to Q is r√2, which must equal 14 - r. Solving:

r√2 = 14 - r

r(√2 + 1) = 14

r = 14/(√2 + 1) = 14(√2 - 1)/( (√2 + 1)(√2 - 1) ) = 14(√2 - 1)/1 = 14√2 - 14

Yes, that's correct. Therefore, the radius is 14(√2 - 1). However, since the problem asks for a common fraction, perhaps there's a different interpretation. Wait, maybe the semicircles are not tangent to both axes but to the main circle and to each other. Let me consider that. Suppose there are two semicircles, each tangent to the main circle and to each other, and each lying in one of the two smaller regions formed by the radii AQ and BQ. Let's model this.

Let the two semicircles be S1 and S2. S1 is in the first quadrant, tangent to AQ (x-axis), S2 is in the second quadrant, tangent to BQ (y-axis). Both S1 and S2 are tangent to the main circle and to each other. The centers of S1 and S2 would be at (r, r) and (-s, s). The distance from (r, r) to Q is sqrt(r^2 + r^2) = r√2, which must equal 14 - r. Similarly for S2, distance from (-s, s) to Q is s√2 = 14 - s. Therefore, r = s = 14/(√2 + 1) = 14(√2 - 1). The distance between the centers (r, r) and (-s, s) is sqrt{(r + s)^2 + (r - s)^2}. Since r = s, this distance is sqrt{(2r)^2 + 0} = 2r. But the semicircles are tangent to each other, so the distance between centers should be equal to r + s = 2r. Therefore, this condition is satisfied. Therefore, the radius is indeed 14(√2 - 1). Therefore, the answer is 14(√2 - 1), which is approximately 5.796. However, since the problem asks for a common fraction, and √2 is irrational, perhaps the answer is to be expressed as 14(√2 - 1), which is an exact value but not a common fraction. Alternatively, perhaps there's a different approach that leads to a fractional answer.

Wait, perhaps the problem refers to the two smaller regions as each being a semicircle whose diameter is along the radii AQ and BQ. Let me consider that. If the diameter of the semicircle is along AQ, then the radius of the semicircle would be half of AQ, which is 7 inches. But the problem states "the two smaller regions are tangent semicircles", so if each semicircle has a diameter of 7 inches, then their radius is 3.5 inches. However, this seems too simple, and the problem mentions "the radius of the smaller semicircle", implying a specific calculation. Also, 3.5 is 7/2, which is a common fraction. Therefore, maybe the answer is 7/2. But how?

Wait, if the semicircles are constructed such that their diameters are along AQ and BQ, then each semicircle would have a radius of 7 inches. But then they are not smaller regions. Alternatively, if the semicircles are constructed within the sector formed by AQ and BQ, each with diameter equal to the radius of the main circle, but that would make their radius 7 inches again. Alternatively, perhaps the semicircles are inscribed in the right angle, each tangent to the main circle and to each other. In that case, the radius would be 14(√2 - 1), which is approximately 5.796, which is less than 7, so smaller. However, this is not a common fraction.

Alternatively, perhaps the problem refers to the two smaller regions as the two semicircular regions formed by the tangents to the main circle and the two radii. Wait, but the problem says "the two smaller regions are tangent semicircles", so each is a semicircle. Therefore, I think the correct radius is 14(√2 - 1), which is approximately 5.796. However, since the problem asks for a common fraction, maybe there's a different approach.

Wait, perhaps the problem is referring to the two smaller regions as the two semicircular regions formed by the tangents from a point. Wait, but the problem mentions "the two smaller regions are tangent semicircles". Therefore, each semicircle is tangent to the other semicircle and to the main circle. Therefore, the radius is 14(√2 - 1). Therefore, the answer is 14(√2 - 1). However, since the problem asks for a common fraction, and √2 is irrational, perhaps the answer is to be expressed as 14(√2 - 1), which is already simplified. Alternatively, maybe the problem expects the answer in terms of fractions multiplied by √2, but that's still not a common fraction. Therefore, perhaps the problem has a typo, and the answer is supposed to be 7/2, but that doesn't make sense geometrically.

Alternatively, perhaps the problem refers to the two smaller regions as each being a semicircle with diameter equal to the radius of the main circle, which would be 7 inches, but that would make the radius 7/2, which is a common fraction. However, this contradicts the geometric reasoning. Therefore, I think the correct answer is 14(√2 - 1), even though it's not a common fraction. Therefore, I will proceed with this answer.

**Final Answer**
\boxed{14(\sqrt{2} - 1)}
</think>