Problem Statement:
Line segment PQ is 6 metres in length and is tangent to the inner circle of the two concentric circles at point R. It is known that the radii of the two circles are integers. Find the radius of the outer circle.

Solution:

Step 1: Labeling the diagram
Let us label the diagram as shown below:

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Step 2: Understanding the problem statement
From the problem statement, we know that:
- PQ is tangent to the inner circle at point R.
- PQ has a length of 6 metres.
- The radii of the two circles are integers.
- We need to find the radius of the outer circle.

Step 3: Using the tangent property
We know that PQ is tangent to the inner circle at point R. This means that PR is perpendicular to PQ. Therefore, we can use the tangent property to find the length of PR.

We know that the tangent to a circle is perpendicular to the radius drawn to the point of contact. Therefore, angle PRQ is a right angle. Also, QR is equal to the radius of the inner circle.

Using Pythagoras theorem, we can find the length of PR as follows:
PR² = PQ² + QR²
PR² = 6² + QR²
PR² = 36 + QR²

Step 4: Finding the radii of the two circles
Let the radius of the inner circle be r. Then, we can express QR as r.

Using the equation derived in Step 3, we can substitute QR as r and simplify:
PR² = 36 + r²
We know that PR is the difference between the radii of the two circles. Let the radius of the outer circle be R. Then, we can express PR as R - r.

Substituting PR as R - r, we get:
(R - r)² = 36 + r²
Expanding the left-hand side, we get:
R² - 2Rr + r² = 36 + r²
Simplifying, we get:
R² - 2Rr = 36
R(R - 2r) = 36

Since R and r are integers, we need to find two factors of 36 such that their difference is even. The only such factors are 6 and 6, which means that R - 2r = 6.

Substituting this value in the equation derived above, we get:
R(6) = 36
R = 6

Step 5: Final answer
Therefore, the radius of the outer circle is 6 metres.