The alternate segment theorem (also known as the tangent-chord theorem) states that in any circle, the angle between a chord and a tangent through one of the end points of the chord is equal to the angle in the alternate segment.
In the above diagram, the angles of the same color are equal to each other. For easily spotting this property of a circle, look out for a triangle with one of its vertices resting on the point of contact of the tangent to the circle. In this wiki we will be learning this theorem in detail along with proofs and its applications in several arenas with the help of examples.
Let's first get familiar with the definition of alternate segment theorem and then we will understand it by a detailed explanation.
Alternate Segment Theorem
In any circle, the angle between a chord and a tangent through one of the endpoints of the chord is equal to the angle in the alternate segment, i.e. the angle subtended by the chord in the opposite side of the previous angle.
Now let's go through the following explanation to have a clear understanding of the theorem:
To be more explicit, consider the above circle with center , with a chord . A tangent passes through the point . Consider any point on on the side of opposite to where is situated. Then we have . For example, if then we have .
What if we arbitrarily move on that side of ? The statement still remains true because the angle subtended by minor arc is constant, irrespective of the position of .
In this section we will prove the Alternate Segment Theorem through two different methods.
Consider the above figure. We've taken diametrically opposite to , so passes through , the center of . Let and . We want to prove that . Notice that since the angle subtended by a diameter is a right angle, we have . Also since the radius through the touch point is perpendicular to the tangent, we have . Now in right angled , we get . And from the figure we can easily see that . Hence .
Note: There is another configuration when the angle at is subtended by the major arc (when is obtuse). This is left to the reader as an exercise.
Here's another proof:
Since and are both the radii of the circle,
Since is isosceles, we have
Since is a tangent, we have
Then from and it follows that
Now, from the properties of an inscribed triangle, we know that
To ensure that you have mastered this theorem, work through the following examples:
In the above diagram it is given that What is
From the tangent-chord theorem, the angle between a tangent and a chord that meet on a circle, is equal to the inscribed angle on the opposite side of the chord. Thus,
In the above diagram find the value of , provided that is .
Since we have
In the above diagram, and What is
According to the tangent-chord theorem, we have
For the sum of its three internal angles must equal Thus, from it follows that
Now it's your turn to solve the following problem:
For deeper insights into the theorem, let's work through some more examples:
In the above diagram, is tangent to circle where the point of contact is If the radius of circle is 10 and what is the length of the arc
Since is a tangent line, implying Then since is an isosceles triangle, This implies
Therefore, the length of the arc denoted by is
The tangents at , to the circumcircle of meet at . The line through parallel to meets at . Prove that .
with transversals and , we have and . Also and are tangents from the same point , so and consequently, . Since there are tangents, we should consider the alternate segment theorem, by which we have . Combining all these results, we findRefer to the figure above. Let's figure out what we know from the problem statement. Since
In particular, , so is cyclic. Hence . So we finally have this long chain
In particular in , so .