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Author:
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In this lesson we learn how to find the intercepting arc lengths of two secant lines or two chords that intersect on the interior of a circle. We also find the angle given the arc lengths.

Background is covered in brief before introducing the terms chord and secant.

Then a formula is presented that we will use to meet this lesson's objectives. The formula is derived in a video and the lesson winds down with two practice examples.

Tutorial

In order to get the most out of this lesson you should already have a good grasp on the following concepts.

The sum of the interior angles of a triangle is 180 degrees.

**a + b + c = 180**

You should also know definition of an *inscribed angle* and understand the notion of an *intercepting arc,* and should be comfortable using the formula that relates them (see this lesson for review)

**a = 1/2 b**

The notion of a supplementary angle and of a linear pair will also be used in this lesson.

Two terms that might be new are **chord** and **secant line**.

A chord is just a line segment conneting two points on a circle.

A **secant line**, or just a secant, is similar to a chord. A secant is a line that intersects a circle at two points, rather than a tangent that only intersects at one point.

Shortly we will derive a formula that applies to a situation like this:

We'd like to know how the angle **a** at the intersection of chords relates to the arcs **B** and **C**. It turns out that the measure of **a** is one half the sum of the measures of **B** and **C**, or **a = (B + C)/2. **We will now derive this formula!

This quick video shows how one can derive the formula relating the angles of secant lines that intersect inside a circle with their intercepting arc lengths on that circle. It assumes knowledge of triangles, complementary angles, and the relationship between inscribed angles and their intercepting arc lengths.

Source: Colin O'Keefe

Now that we understand the relationship between interior intersections and their intercepting arcs,lets try some applications.

1. Given the lengths of intercepting arcs, determine the angle of intersection:

Solution:

Here we can simply apply the formula.

a = (70 + 40)/2

a = 110/2

a = 55

2. Given the angle and one arc length, determine the other arc length:

Solution:

This solution requires a little more algebra, but is still pretty straightforward.

Our angle is 80 and our known length is 100, pluging these into the formula we get

80 = (x + 100)/2

Multiplying both sides by 2 gives

160 = x + 100

and solving for x we have

x = 60