Question

Point upper D lies on the circumference of the circle opposite to point upper E. A line segment joins upper D with upper E and passes through center point upper C. Point upper F and upper G are opposite each other on the circumference, to the left of point C. Chord upper F upper G is perpendicular to line segment upper D upper E, as indicated by a right angle marked at the point of intersection which is labeled as point upper H. A line segment joins point upper F with point upper C.

If the radius of circle C
is 8 mm and m∠FCE=60°
, then find the exact measure of FG¯¯¯¯¯¯¯¯
.

(1 point)
FG=

mm

Answer 1

Put C at the origin and let CE be the diameter along the negative x-axis (so E = (−8,0)). Since ∠FCE = 60°, the radius CF makes a 60° angle with CE, so CF points at 180° − 60° = 120°. Thus
F = (8 cos 120°, 8 sin 120°) = (−4, 4√3).

FG is perpendicular to the diameter CE, so it is the vertical chord through x = −4. Its endpoints are at (−4, ±4√3), so
FG = 2·4√3 = 8√3 mm.

Answer: FG = 8√3 mm.