In this lesson we learn and apply the formula that relates arc lengths to the angle of lines that intersect outside of a circle.
Background is briefly reviewed and the formula is introduced. The derivation of the formula is presented in video format before some example exercises are worked.
In order to get the most out of this lesson, you ought to be comfortable with the following facts and concepts.
1. The sum of the angles of a triangle is 180 degrees
a + b + c = 180
2. The concept of a supplementary angle
3. The definition of an inscribed angle, and the formula that relates inscribed angles with the lengths of their intercepting arcs. For review, see this lesson.
a = 1/2 b
4. The defnintions of secants and tangents
In this lesson we will be look at situations like the following
secants intersecting outside a circle
tangents intersecting outside the circle
Or even in a case like this, where tangents and secants intersect outside the circle
In all three cases, it turns out that the values a, x and y are related by the formula
a = 1/2(y - x)
Now we'll derive this formula!
This video derives the formula relating secants whose intersection falls outside of a circle with the lengths of the arcs the secants intersect.
Source: Colin O'Keefe
Now that we understand the formula for each of the three cases we can use it to work through some examples.
1. Find the arc length y
Well, using the formula we know that the angle is half the difference of the arc lengths, or
60 = (y - 40) / 2
multiplying both sides by 2 gives
120 = y - 40
and adding 40 to both sides we get
160 = y
2. Determin the arc length x from the information given.
First we need to determine the measure of the angle at the intersection of the secants.
We notice that it is the third angle in a triangle whose other two angles are 60 and 90
form this we deduce our angle
180 - (60 + 90) = 30
Next, we use the formula to determine x, using our angle measure and the known arc length:
30 = (100 - x)/2
multiply both sides by 2
60 = 100 - x
add x to both sides and subract 60 from both sides
x = 40
3. find the angle from the arc lengths
First we need to determine the other arc length.
We notice that both lines are tangent, so that the other arc is just the rest of the circle.
Hence the value is 360 - 210 = 150
Now we can use our formula
a = (210 - 150) / 2
a = 60 / 2
a = 30