Question

given the circle below with chords \\( \overline{cd} \\) and \\( \overline{ef} \\). find the length of \\( \overline{eg} \\). round to the nearest tenth if necessary.

Explanation:

Step1: Apply chord intersection theorem

If two chords intersect at a point inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord. Let $EG = x$. Then $FG = EF - EG$, but first use the theorem:
$$CG \times GD = EG \times GF$$
We know $CG=39$, $GD=23$, $GF=43$, so substitute these values:
$$39 \times 23 = x \times 43$$

Step2: Calculate left-hand side

Compute the product of 39 and 23:
$$39 \times 23 = 897$$

Step3: Solve for $x$

Rearrange the equation to isolate $x$:
$$x = \frac{897}{43}$$

Step4: Compute and round the value

Calculate the division and round to the nearest tenth:
$$x \approx 20.9$$

Answer:

20.9