9.10. Segments from Secants http://www.ck12.org
If we draw in the intersecting chords, we will have two similar triangles.
From the inscribed angles and the Reflexive Property(^6 R∼=^6 R), 4 PRS∼4T RQ. Because the two triangles are
similar, we can set up a proportion between the corresponding sides. Then, cross-multiply.c+ad=a+cb⇒a(a+b) =
c(c+d)
Two Secants Segments Theorem:If two secants are drawn from a common point outside a circle and the segments
are labeled as above, thena(a+b) =c(c+d). In other words, the product of the outer segment and the whole of
one secant is equal to the product of the outer segment and the whole of the other secant.
Example A
Find the value of the missing variable.
Use the Two Secants Segments Theorem to set up an equation. For both secants, you multiply the outer portion of
the secant by the whole.
18 ·( 18 +x) = 16 ·( 16 + 24 )
324 + 18 x= 256 + 384
18 x= 316x= 17