Intersecting chords theorem
A chord is defined as a segment that has both endpoints in the circumference of the circle.
In the provided figure there are two chords: CD and AB
When two chords intersect they define a point (called E).
The intersecting chords theorem states that the product of the four segments defined at the intersection point E satisfies the relation:
CE * ED = AE * EB
We are given the values of CE=12, ED=3x-11, AE=x-1, and EB=30
Substituting in the above relation:
12 * ( 3x - 11 ) = (x - 1) * 30
36x - 132 = 30x - 30
Rearranging the equation:
36x - 30x = 132 - 30
6x = 102
Dividing by 6:
x = 102/6
x = 17
We are required to calculate CD, the total length of the segment from C to D:
CD = CE + ED = 12 + 3(17) - 11 = 12 + 51 - 11 = 52
Answer: CD = 52