http://www.ck12.org Chapter 9. Circles
9.10 Segments from Secants
Here you’ll learn how to solve for missing segments from secants intersecting circles.
What if you wanted to figure out the distance from the orbiting moon to different locations on Earth? At a particular
time, the moon is 238,857 miles from Beijing, China. On the same line, Yukon is 12,451 miles from Beijing.
Drawing another line from the moon to Cape Horn (the southernmost point of South America), we see that Jakarta,
Indonesia is collinear. If the distance from Cape Horn to Jakarta is 9849 miles, what is the distance from the moon
to Jakarta? After completing this Concept, you’ll be able to solve problems like this.
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In addition to forming an angle outside of a circle, the circle can divide the secants into segments that are proportional
with each other.
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.