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Home » Circles » Inscribed Shapes » Kite Inscribed in a Circle

Kite Inscribed in a Circle

Last updated: Mar 27, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

When we inscribe a kite is in a circle, all four of the kite's vertices lie on the circle's circumference.

In today's lesson, we will show that in the case of a kite inscribed in a circle, the axis of symmetry of the kite is the circle's diameter.

This is a pretty straightforward geometry proof, so today's lesson is going to be rather short.

Problem

ABCD is a kite that is inscribed in circle O. Show that AC is the diameter of the circle.

kite in a circle

Strategy

We have previously discussed which quadrangles can be inscribed in a circle, and we have shown that such quadrangles have opposing angles that are supplementary. Meaning, they add up to 180°.

In addition, we also know that in a kite, the opposing angles on either side of the axis of symmetry are equal.

Now let's go ahead and put these two facts together. We see that this kite's opposing angles are both supplementary and equal, which means they must be 90° each. And by the inscribed angle theorem, they subtend an arc that is 180° - and thus the chord of that arc is the diameter.

Proof

Axis of symmetry of a kite inscribed in a circle:

(1) m∠ABC + m∠CDA= 180° //Opposing angles of an inscribed quadrangle are supplementary
(2) m∠ABC = m∠CDA //Opposing angles on either side of a kite's axis of symmetry are equal.
(3) m∠ABC = m∠CDA =90° //(1), (2), Transitive property of equality and algebra.
(4) Arc(ABC)=Arc(CDA) = 180° //(3), Inscribed angle theorem
(5) AC is the diameter of O //Definition of diameter

« Parallel Lines Drawn Through a Point on the Diagonal of a Parallelogram
Finding the Ratio of Triangle Areas With the Same Base »

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
  • Polygons
  • Quadrangles
    • Kites (Deltoids)
    • Parallelograms
    • Rectangles
    • Rhombus
    • Squares
    • Trapezoids
  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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