Circles: Special Segments in Circles

Properties of Secants

The lengths of segments of secants are related in a particular way when the secants intersect outside the circle.

Take a look at the circle and the secants drawn.

Circle with secant PB which intersects the circle at point A; secant PY which intersects the circle at point X
Circle with intersecting secant lines

The product of PA times PB equals the product of PX times PY.

PA • PB = PX • PY

 

Example:

What is XY in the figure below?

Circle with secant PB which intersects the circle at point A, PA equals 6 and AB equals 4; secant PY which intersects the circle at point X, PX equals 5
Circle with intersecting secant lines

Solution:

PA = 6
PB = 6 + 4 = 10
PX = 5
PY = 5 + XY

PA • PB = PX • PY
6(10) = 5(5 + XY)
60 = 25 + 5XY
35 = 5XY

7 = XY