You know that the hypotenuse of the triangle shown is equal to the radius of the circle, 12. You also know that the longer leg of the triangle shown is half of 19.2, so is 9.6. You can use the Pythagorean theorem to find x from these values.
x² + 9.6² = 12²
x² = 144 - 92.16 = 51.84
x = √51.84 = 7.2
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You can make an educated guess based on your knowledge of right triangles. Since the long leg of the right triangle is longer than 6√2 ≈ 8.6, x must be shorter than 8.6. (The longest that two equal sides will be is 6√2—making an isosceles right triangle. When the sides are unequal, one will be longer than that and the other will be shorter.) This observation leaves only the choices 7.2 and 4.8.
You also know that (5, 12, 13) is a Pythagorean triple. Compared to (x, 9.6, 12) we note that the ratio of 9.6:12 is somewhat less than the ratio 12:13. Thus the short side, x, must have a ratio to 12 that is somewhat larger than the ratio 5:13. (As one leg of a right triangle gets shorter, the other must get longer for the same hypotenuse.) This observation eliminates 4.8 as a possible answer, leaving 7.2 as a good guess.
You can also estimate x based on the figure, assuming it is drawn nearly to scale. Often, that is a bad choice, as the figures are rarely drawn to scale, and are often drawn so they are purposely misleading. Here, it appears the figure may be fairly accurate. (Attached is a figure that IS accurately drawn to scale.)