Intersecting chords theorem - Wikipedia
Let P be the point where the two chords (and a diameter) meet. The Power of a Point principle says that every chord through a particular point of a circle is divided into sub-segments such that the product of the lengths of. Intersecting Chords Theorem: If two chords intersect inside a circle so that one is divided into segments of length and and the other into. In Figure 1, circle O has radii OA, OB, OC and OD If chords AB and CD are of equal Figure 4 In a circle, the relationship between two chords being equal in.
There are some fun problems in probability theory that involve intersections of line segments. One is "What is the probability that two randomly chosen chords of a circle intersect? An exact answer For this problem, a "random chord" is defined as the line segment that joins two points chosen at random with uniform probability on the circle.
The probability that two random chords intersect can be derived by using a simple counting argument. Suppose that you pick four points at random on the circle. Label the points according to their polar angle as p1, p2, p3, and p4. As illustrated by the following graphic, the points are arranged on the circle in one of the following three ways.
Parallel Chords Problem
A simulation in SAS You can create a simulation to estimate the probability that two random chords intersect. It converts those points to x,y coordinates on the unit circle.
It then computes whether the chord between the first two points intersects the chord between the third and fourth points. The line may be a tangent, touching the circle at just one point. The line may miss the circle entirely.
- Intersecting Chords Theorem
- Advanced information about circles
- Arcs and Chords
The point where a tangent touches a circle is called a point of contact. It is not immediately obvious how to draw a tangent at a particular point on a circle, or even whether there may be more than one tangent at that point. Theorem Let T be a point on a circle with centre O.
Proof First we prove parts a and c. Let be the line through T perpendicular to the radius OT. Let P be any other point onand join the interval OP.
Angles of intersecting chords theorem
Hence P lies outside the circle, and not on it. This proves that the line is a tangent, because it meets the circle only at T.
It also proves that every point onexcept for T, lies outside the circle.Intersecting Chords
It remains to prove part b, that there is no other tangent to the circle at T. Let t be a tangent at T, and suppose, by way of contradiction, that t were not perpendicular to OT. Hence U also lies on the circle, contradicting the fact that t is a tangent.
Tangents from an external point have equal length It is also a simple consequence of the radius-and-tangent theorem that the two tangents PT and PU have equal length.