Secants P N and L N intersect at point N outside of the circle. Secant P N intersects the circle at point Q and secant L N intersects the circle at point M. The length of P N is 32, the length of Q N is x, the length of L M is 22, and the length of M N is 14. In the diagram, the length of the external portion of the secant segment PN is . The length of the entire secant segment LN is . The value of x is .

In the diagram, the length of the external portion of the secant segment PN is x. The length of the entire segment LN is 36. The value of x is 17.75.

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ap8997154 ap8997154 · Answer 2 · 2022-01-28T06:37:05+01:00

To solve the problem we should know about Intersecting Secant Theorem.

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 is equal to the product of the outside segment of the other secant and its length.

a(a+b)=c(c+d)

The length of the line segment is 15.75.

Given to us

PN = 32
QN = x
LM = 22
MN = 14

Using the Intersecting Secant theorem

a(a+b) = c(c+d)

QN(PQ+QM) = MN(LM+MN)

QN(PN) = MN(LM+MN)

[tex]x(32) = 14(22+14)\\\\32x = 14(36)\\\\x=\dfrac{14 \times 36}{32}\\\\x = 15.75[/tex]

Hence, the length of the line segment is 15.75.

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