Concentric Circle Theorem
Concentric Circle Theorem states that, “If the chord of outer circle touches the inner circle at one point, the chord is bisected at the point of contact.”
Proof:
Consider two concentric circles with center O and radii r (larger circle) and R (smaller circle).
Draw a chord AB of the larger circle that touches the smaller circle at point C. Mark the center of the circles as O.
Now, connect OC, which is the radius of the smaller circle.
In △OAC and △OBC:
OA = OB (Both are radii of the larger circle)
OC = OC (Common side, the radius of the smaller circle)
∠OAC = ∠OBC = 90° (Perpendicular to the tangent at the point of contact is 90 degrees)
So, △OAC and △OBC are congruent by the hypotenuse-leg congruence criterion.
By the congruence of triangles, AC = BC. Therefore, the chord AB is bisected at point C, where it touches the smaller circle.
This phenomenon occurs because the tangent to a circle is perpendicular to the radius at the point of contact. Hence, the two segments of the chord from the point of contact to the ends of the chord are equal in length.
Concentric Circles
Concentric circles are defined as two or more circles that share the same center point, known as the midpoint, but each has a different radius. If circles overlap yet have different centers, they do not qualify as concentric circles. According to Euclidean Geometry, two concentric circles must have two different radii. The space between the circumference of these two circles is called the annulus of a circle.
In this article, we will learn about concentric circles, the theorem on concentric circles, the region between the concentric circles, Concentric Circle Equations, and Concentric Circles examples in detail.
Table of Content
- What are Concentric Circles?
- Concentric Circles Meaning
- Concentric Circle Examples
- Region between Two Concentric Circles
- Concentric Circle Theorem
- Concentric Circle Equations
- Solved Examples on Concentric Circles
- Practice Questions on Concentric Circles