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