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Home » Quadrangles » Rhombus » A Parallelogram with Perpendicular Diagonals is a Rhombus

A Parallelogram with Perpendicular Diagonals is a Rhombus

Last updated: Oct 29, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

A rhombus is a special kind of parallelogram, in which all the sides are equal. We've seen that one of the properties of a rhombus is that its diagonals are perpendicular to each other. Here we will show the converse- that if a parallelogram has perpendicular diagonals, it is a rhombus - all its sides are equal.

Problem

ABCD is a parallelogram, and its diagonals are perpendicular - AC⊥DB. Show that ABCD is a rhombus.

rhombus with diagonals

Strategy

ABCD is a rhombus if all its sides are equal. We know that the opposite sides in ABCD are equal, because it is a parallelogram. So we need to show that one pair of adjacent sides is equal. We will do this using congruent triangles.

Let's take triangles ΔAOD and  ΔCOD. We have one common side - OD, and since the diagonals are perpendicular, m∠AOD = m∠COD=90°, and so ∠AOD ≅ ∠COD. We now need either another side, or another angle to show the triangles are congruent.

So let's review the properties of parallelograms - one of them is that the diagonals bisect each other. So |AO|=|OC|, and we have our second side for the Side-Angle-Side postulate.

And if triangles ΔAOD and  ΔCOD are congruent, |AD|=|DC|, and we are done showing that a parallelogram with perpendicular diagonals is a rhombus.

Proof

(1) |AD|=|BC| ; |AB|=|DC| //Opposite sides of a parallelogram are equal
(2) |OD|=|OD| //Common side, reflexive property of equality
(3) AC⊥DB //Given
(4) m∠AOD = m∠COD=90° // (3), definition of perpendicular lines
(5) ∠AOD ≅ ∠COD // (4), definition of congruent angles
(6) |AO|=|OC| // diagonals of a parallelogram bisect each other
(7) ΔAOD≅ΔCO // (2), (5), (6). Side-Angle-Side postulate
(8) |AD|=|DC| // Corresponding sides in congruent triangles (CPCTC)
(9) |AD|=|BC|=|DC|=|AB| // (1),(8)
(10) ABCD is a rhombus //(9), defintion of a rhombus (parallelogram with equal sides)

« Heron's Formula
Area of a Right 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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