Examine the following diagram, where AP¯¯¯¯¯¯¯¯
is secant to the circle at points A
and B,
and CP¯¯¯¯¯¯¯¯
is secant to the circle at points C
and D.
The lines intersect outside the circle at point P.
A circle, with no center shown, and two secants, segment A P and segment C P, as described in the text. Segment A B equals 8, segment B P equals 10, and segment C P equals 20. What is the length of PD¯¯¯¯¯¯¯¯?
Responses
12
12
7
7
9
9
11
We can use the Secant-Secant Power Theorem to solve this problem. According to the theorem, when two secants intersect outside a circle, the product of the whole secant and its external segment is equal to the product of the other secant and its external segment. In equation form:
(AP)(BP) = (CP)(DP)
Substitute the given values into the equation:
(8)(10) = (20)(DP)
80 = 20DP
DP = 4
Therefore, the length of segment PD is 4.