To solve the problems:

Problem 1:

The formula to calculate the distance between the centers of two circles when the length of their external common tangent is known is:

$d = \sqrt{L^2 + (r_1 + r_2)^2}$

Where:

• $L = 24$ cm (length of the external common tangent),
• $r_1 = 14$ cm (radius of the first circle),
• $r_2 = 4$ cm (radius of the second circle).

Substitute the values:

$d = \sqrt{24^2 + (14 + 4)^2}$

$d = \sqrt{576 + 324}$

$d = \sqrt{900} = 30 \, \text{cm}.$

So the answer is not listed; double-check external tangent data.

Problem 2:

The formula for the internal common tangent is:

$L = \sqrt{d^2 - (r_1 - r_2)^2}$

Where:

• $d = 29$ cm (distance between centers),
• $r_1 = 13$ cm,
• $r_2 = 7$ cm.

Substitute the values:

$L = \sqrt{29^2 - (13 - 7)^2}$

$L = \sqrt{841 - 36}$

[ L = \sqrt{805} \approx 28,2

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