Answer:
56 cm
Explanation:
The tangents from the 90° angle will form a square with the radii that has a side length of 4. If we call the length of the short side of the right triangle "x", then the tangent lengths are ...
on the short side of the triangle: 4, x-4
on the hypotenuse side of the triangle: x-4, 24-(x-4) = 28-x
on the long side of the triangle: 4, 28-x
The perimeter is twice the sum of the unique tangent lengths:
P = 2(4 + (x-4) + (28-x)) = 2·28
P = 56 . . . . . the perimeter is 56 cm.
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Using the Pythagorean theorem on side lengths x and 32-x and hypotenuse 24, we find x = 16-4√2 ≈ 10.34, the length of the short side (in cm).