Answer:
x = 4 . . . cm
Explanation:
You want the measure of the segment CE from external point C to circle O, given several other dimensions of chords and segments in the circle.
Crossing chords
As in the attachment, we can extend segment DO across the circle to make diameter DX. Then we can use the relation for chord lengths to find the radius of the circle (r).
Segments of crossing chords have the same product:
FA·FE = FD·FX
7·4 = 2·(2r -2) . . . . . use given segment values
14 = 2r -2 . . . . . . divide by 2
16 = 2r . . . . . . . add 2
r = 8 . . . . . . . divide by 2
The radius of the circle is 8 cm.
Secants
We can also extend radius EO across the circle to make diameter EY, as shown in the attachment.
The relation involving the product of secant segments can be used to find x:
CE·CY = CB·CA
x(x +16) = 5(5+4+7)
x² +16x = 80 . . . . . . . . simplify
x² +16x +64 = 144 . . . . . . . add 64 to complete the square
(x +8)² = 12² . . . . . . . . . write as squares
x +8 = 12 . . . . . . . . positive square root
x = 4 . . . . . . . . subtract 8
The value of x is 4.
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Additional comment
The attachment is accurately drawn.