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Arc Measures: Secants And Chords With Interior Intersection

Arc Measures: Secants And Chords With Interior Intersection

Author: c o

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.

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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.

New terms and concepts

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!

Deriving the Formula: intersecting secants and intercepting arcs

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

Some applications

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:


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:


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