Intersecting Secant Theorem helps to establish the relation between two secants of a circle. The length of the side AE will be equal to 3 units.
What is Intersecting Secant Theorem?
When two line secants of a circle intersect each other outside the circle, the circle divides the secants into two segments such that the product of the outside segment and the length of the secant are equal to the product of the outside segment and the other secant's length.
a(a+b)=c(c+d)
If we look into the triangle BOH, then the length of the side BH using the Pythagoras theorem can be written as,
[tex]OB^2=OH^2+BH^2\\\\10^2 = 6^2 + BH^2\\\\BH= 8\rm\ units[/tex]
Now, the length of BC will be twice the length of BH, therefore, the length of the line BC can be written as,
[tex]\text{Length of BC} = 2 \times (\text{Length of BH})\\\\\text{Length of BC} = 2 \times 8 = 16\rm\ units[/tex]
Now, using the Intersecting secant theorem the length of AE can be written as,
[tex](AB \times AC) = (AE \times FA)\\\\2(2+16) = AE(AE+9)\\\\36=AE^2+9AE\\\\AE^2+9AE-36 = 0\\\\AE^2+12 AE - 3AE -36=0\\\\AE(AE+12)-3(AE+12)=0\\\\(AE-3)(AE+12)=0\\\\AE = 3, -12[/tex]
Since the length can not be negative, therefore, the length of the side AE will be equal to 3 units.
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