Question

The closed points and lines are labeled and placed as follows. The center of the circle is labeled as closed point upper A. Neither line passes through it. Closed point upper B is on the top right edge of the circle, at approximately the top one-fourth of the circle. Closed point upper C is on the left of the circle, directly across from upper B. A horizontal line begins outside the circle and passes through upper C and upper B and continues to extend to the right, where it intersects with the second line. The point of intersection is labeled upper F. Closed point upper D is on the lower left edge of the circle, almost at the bottom. Closed point upper E is on the right edge of the circle, just below the center of the circle. A diagonal line extends upward, left to right, connecting upper D and upper E, and continues on to intersect with the other line at closed point upper F.

Identify and use the lengths of radii, diameters, secants, and chords within the given circle to solve for the measure of CF¯¯¯¯¯¯¯¯
if DE=11
, EF=7
, and BF=4
.

(1 point)
Responses

CF=27.5

upper C upper F equals 27.5

CF=23.5

upper C upper F equals 23.5

CF=35.5

upper C upper F equals 35.5

CF=31.5

upper C upper F equals 31.5
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Answer 1

Use the secant-secant theorem from external point F:
BF(BF + BC) = EF(EF + ED).

Compute the right side: 7(7 + 11) = 7·18 = 126.
So 4(4 + BC) = 126 → 4 + BC = 31.5 → BC = 27.5.

CF = CB + BF = 27.5 + 4 = 31.5.

Answer: CF = 31.5.