Using this property: if two chords of a circle intersect, the product of the sections of one of the chords equals
the product of the sections of the other.
Since EF = 15 and EU = 8, UF = 7.
Since GH = 21 and GV = 5, HV = 16.
On chord XUVY, call section XU = x and VY = y.
Using chord EF and chord XUVY: 7·8 = x(y + 6).
Using chord GH an chord XUVY: 5·16 = y(x + 6)
Using 5·16 = y(x + 6) ---> 80 = y(x + 6) ---> y = 80 / (x + 6)
Using 7·8 = x(y + 6) ---> 56 = xy + 6x ---> xy = 56 - 6x ---> y = (56 - 6x) / x
Setting these two equations equal to each other: 80 / (x + 6) = (56 - 6x) / x
Cross-multiplying: (56 - 6x)(x + 6) = 80x
Simplifying: x2 + 10x - 56 = 0
(x + 14)(x - 4) = 0
x = 4
From here, you can calculate both the value of y and the full length of the chord.