Secants: In and Out

** **

**By**

** **

**Kiisha Gibbs**

**Problem:**

Explore
how the angle formed by two secants of a circle relates to the intercepted
arc(s).

There are three
situations that we explored:

1) the secants intersect on the circle (vertex
of the angle lies on the circle);

2) the secants intersect inside the circle;

3) the secants intersect outside the circle.

For each case is
simple discussion relating the measure of angle ABC and the intercepted arc(s).

FIGURE1

FIGURE2

FIGURE3

In exploring the location
of point A, where the secants intersect, I discovered that there was a
“number line” effect :

Negative
(subtract) Zero Positive (add)

When A is outside
the circle, you subtract the intercepted arcs and take half, when A is on the
circle you just take half of the intercepted arc and when A is inside the
circle you add the intercepted arcs and take half.

Click here to view the GSP file where you can further
explore the location of A.