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Home » Quadrangles » Kites (Deltoids) » The Diagonals of a Kite are Perpendicular to Each Other

The Diagonals of a Kite are Perpendicular to Each Other

Last updated: Sep 30, 2019 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

We have already shown that the diagonal that connects the two corners formed by the sides that are equal bisects the angles at those corners. So it is now easy to show another property of the diagonals of kites- they are perpendicular to each other.

Problem

ABCD is a kite. Show that the diagonals are perpendicular, that is, AC⊥DB.

diagonals of kite

Strategy

We will follow the exact same strategy as we did to prove a very similar theorem - that the Diagonals of a rhombus are perpendicular to each other. And we will use triangle congruency.

In addition, we will also use what we've earlier proved for kites- that the diagonal that connects the two corners formed by the sides that are equal bisects the angles at those corners.

Using the above, we can show that triangles  ΔAOD and  ΔAOB are congruent using the Side-Angle-Side postulate, and from that that ∠AOD ≅ ∠AOB.

And now, since ∠AOD and ∠AOB are a linear pair, we use the Linear Pair Perpendicular Theorem – If two straight lines intersect at a point and form a linear pair of equal angles, they are perpendicular.

Proof

(1) ABCD is a Kite //Given
(2) AB=AD                     //(1) definition of a kite
(3) AO=AO                    //Common side, reflexive property of equality
(4) ∠BAC ≅ ∠DAC // (1), in a kite the axis of symmetry bisects the angles at those corners
(5) △AOD≅△AOB      //Side-Angle-Side postulate.
(6) ∠AOD ≅ ∠AOB      //Corresponding angles in congruent triangles (CPCTC)
(7) AC⊥DB     //Linear Pair Perpendicular Theorem

Using this property, it is easy to find a formula for the area of a kite, using the lengths of its diagonals.

« Common Chord of Two Circles
Finding the Length of a Common Chord »

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

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