Answer:
Let's call the center of the larger circle point A and the center of the smaller circle point B. We draw a line connecting A and B, which has a length of 117 cm.
We also draw radii from each center to the points where the tangent line intersects each circle. Let's call the intersection point on the larger circle C and the intersection point on the smaller circle D.
Now, we have a right triangle ABC with hypotenuse AB = 117 cm and legs AC = 35 cm and BC = 10 cm. We can use the Pythagorean theorem to solve for the length of AB:
AB^2 = AC^2 + BC^2
AB^2 = 35^2 + 10^2
AB^2 = 1365
AB ≈ 37 cm
Next, we draw a perpendicular line from point C to line AB, which we'll call point E. Since triangle CDE is also a right triangle, we can use the Pythagorean theorem again to solve for the length of DE:
DE^2 = DC^2 - CE^2
DE^2 = (35-10)^2 - (AB/2)^2
DE^2 = 625 - 18.5^2
DE ≈ 29.7 cm
Finally, we can use the Pythagorean theorem one more time to find the length of the tangent line:
Tangent^2 = TE^2 - TD^2
Tangent^2 = DE^2 - 10^2
Tangent^2 = 29.7^2 - 10^2
Tangent ≈ 28.8 cm
Therefore, the length of the common internal tangent is approximately 28.8 cm.