Question

given the circle below with chords \\(\overline{tu}\\) and \\(\overline{vw}\\). find the length of \\(\overline{ux}\\). round to the nearest tenth if necessary.

Explanation:

Step1: Apply chord intersection theorem

When two chords intersect in a circle, the products of the lengths of their segments are equal. Let $UX = x$. Then:
$$x \times 16 = 19 \times 14$$

Step2: Calculate the product on the right

Compute $19 \times 14$:
$$19 \times 14 = 266$$

Step3: Solve for $x$

Rearrange to isolate $x$:
$$x = \frac{266}{16}$$

Step4: Simplify and round

Calculate the value and round to the nearest tenth:
$$x = 16.625 \approx 16.6$$

Answer:

16.6