Step-by-step explanation:
if two secant segments are drawn to a circle from an exterior point, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment.
in terms of this example :
AP × BP = CP × PD
(5+7) × 7 = (14 + PD) × PD
12×7 = 14×PD + PD²
84 = 14×PD + PD²
PD² + 14×PD - 84 = 0
this quadratic equation can be solved by this generic formula :
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
x = PD
a = 1
b = 14
c = -84
PD = (-14 ± sqrt(196 - 4×1×-84))/(2×1) =
= (-14 ± sqrt(196 + 336))/2 = (-14 ± sqrt(532))/2 =
= (-14 ± sqrt(4×133))/2 = (-14 ± 2×sqrt(133))/2 =
= -7 ± sqrt(133)
PD1 = -7 + sqrt(133) = 4.532562595...
PD2 = -7 - sqrt(133) that is negative and does not make sense for a distance.
I don't know the format you need to have the answer in,
but it is in value
4.532562595... or -7 + sqrt(133)
