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Home » Quadrangles » Rhombus » Diagonals of a Rhombus are Perpendicular to Each Other

Diagonals of a Rhombus are Perpendicular to Each Other

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

We have shown that in a rhombus the diagonals bisect the angles, using triangle congruency. We can follow the same procedure to prove that the diagonals of a rhombus are perpendicular to each other

Problem

In a rhombus ABCD, prove that the diagonals are perpendicular to each other. i.e prove that AC⊥DB.

rhombus with perpendicular diagonals

Strategy

To prove that two lines are perpendicular, when all we have are those two lines, we can 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.

Our diagonals intersect at point O, so we'd need to show the two linear angles formed at that intersection point are equal, and we can do that with triangle congruency.

A rhombus is a parallelogram, so we will use what we already know about parallelograms - that the diagonals bisect each other.

It is then easy to show that the triangles  ΔAOD and  ΔAOB are congruent using the Side-Side-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 rhombus //Given
(2) AB=AD                     //definition of rhombus
(3) AO=AO                    //Common side, reflexive property of equality
(4) BO=OD // A rhombus is a parallelogram, a parallelogram's diagonals bisect each other
(5) △AOD≅△AOB   //Side-Side-Side postulate.     
(6) ∠AOD ≅ ∠AOB     //Corresponding angles in congruent triangles (CPCTC)
(7) AC⊥DB     //Linear Pair Perpendicular Theorem

The converse of this is also true: if a parallelogram's diagonals are perpendicular, it is a rhombus.

« Converse Triangle Midsegment Theorem
Radius of a Circle with an Inscribed Triangle »

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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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…

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