Two circles with radii r₁ and r₂ touch each other internally. If the length of their common tangent is T, which of the following is the correct relationship between T, r₁, and r₂?

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**Solution**

1. For two circles touching externally the center distance is $d=r_1+r_2$.
2. The length of the common external tangent (between the tangent points) is $T=\sqrt{d^2-(r_1-r_2)^2}$.
3. Substitute $d=r_1+r_2$: $T=\sqrt{(r_1+r_2)^2-(r_1-r_2)^2}=\sqrt{4r_1r_2}=2\sqrt{r_1r_2}$.
4. (If they touched internally the only common tangent is at the touch point, giving zero finite length.)

FINAL_ANSWER: d

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