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Home » Circles » Chords » Concentric Circles Intersected by a Secant

Concentric Circles Intersected by a Secant

Last updated: Jan 4, 2020 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

Two circles that have the same center point are called concentric circles. A secant is a line that interest a circle (or any other curved line) at two or more point. We will now show that a secant line that intersects both of the concentric circles creates two congruent segments between the two circles.

Problem

Two congruent circles with center at point O are intersected by a secant. Prove that AB=CD.

concentric circles with secant

Strategy

The secant creates two chords. In the outer circle, it forms chord AD. In the inner circle, it forms chord BC (which is a sub-segment of chord AD). What do we know about chords? One property is that a diameter or radius that is perpendicular to a chord bisects that chord.

concentric circles with secant and radius

In our case, since the two circle are concentric, they have the same center point O. So if we draw such a radius from the center point O, it will bisect both AD and BC - so BE=EC and AE=ED. And with some simple math, we can show that the non-overlapping difference between the line segments is also equal.

Proof

(1) OE⊥BC //Construction
(2) BE=EC //Perpendicular radius bisects the chord
(3) AE=ED //Perpendicular radius bisects the chord
(4)AE-BE=ED-EC //(2), (3) , Subtraction property of equality
(5)AB=CD //Segment addition postulate

« The Axis of Symmetry of a Deltoid Bisects the Other Diagonal
Area of Similar Triangles »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

  • Area of Geometric Shapes
  • Circles
    • Arcs, Angles, and Sectors
    • Chords
    • Inscribed Shapes
    • Tangent Lines
  • Lines and Angles
    • Intersecting Lines and Angles
    • Parallel Lines
    • Perpendicular lines
  • Pentagons and Hexagons
  • Perimeter of Geometric Shapes
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    • Parallelograms
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  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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