CIRCLE INVESTIGATION: FINDING THE RELATIONSHIP OF SECANTS, ANGLES AND ARCS – One Case

By

TERESA L. COX

This write-up investigates the relationship of the
secants, angles, and arcs of a circle when two secants intersect OUTSIDE the
circle. Other cases will be
explored in a later write-up.

First, draw a circle with two secants that intersect
outside the circle. Here is a GSP
picture.

We are curious about the relationship of ÐEFD
and the arcs that are created by the secants FE and FD. Is there a relationship between them?
Is there a mathematical relationship between ÐEFD,
arc GIH, and arc EJD?

At this stage, we know about some relationships
about a circle. Basic facts, such
as the diameter is twice the radius, won’t help us here. The secants do not go through the
center A of the circle. We do know
about inscribed angles (see note below), but these secants go outside the
circle and do not lie on the circle.

NOTE:
The measurement of an inscribed angle of a circle is 1/2 the measurement
of the arc opposite the angle.

As a genus once told me, “Act like a freshman
in geometry and connect any points not already connected.” I want to connect segment HE and create
at least one triangle. Here’s the updated picture, with a segment HE and
new angles ÐFEH and ÐEHD
named Ð1 andf Ð2
respectively. I will also refer to
arc GIH and arc EJD as arc #1 and arc #2 respectively.

We know that inscribed angle Ð1
is 1/2 the measurement of arc #1.
We also know that incscribed angle Ð2
is 1/2 the measurement of arc #2. Ð2
is an exterior angle to the triangle EFH, so we know that Ð2
= Ð1 + ÐF. Let’s restate all these
relationships in algebraic form.

Ð2 = Ð1
+ ÐF

Ð1 = 1/2 arc #1

Ð2 = 1/2 arc #2

Let’s restate the first equation to solve for ÐF.

ÐF = Ð2
- Ð1

Now substitute what we know about the inscribed angles.

ÐF = 1/2 arc #2 – 1/2 arc #1

Factoring out the 1/;2 to simplify,

ÐF = 1/2 (arc #2 – arc #1)

This is the relationship that exists between the
angle and arcs created by the secants of a circle that intersect outside the
circle.

Now,
what do you think happens when the secants intersect INSIDE the
circle? That’s another
writeup!!!!!!

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