Difference between revisions of "Rhombus"

Revision as of 03:27, 31 October 2006

A rhombus is a geometric figure that lies in a plane. It is defined as a quadrilateral all of whose sides are congruent. It is a special type of parallelogram, and its properties (aside from those properties of parallelograms) include:

Its diagonals divide the figure into 4 congruent triangles.
Its diagonals are perpendicular bisectors of eachother.
If all of a rhombus' angles are right angles, then the rhombus is a square.

Proofs

This article would be greatly enhanced by these proofs:

Proof that a rhombus is a parallelogram

All sides of a rhombus are congruent, so opposite sides are congruent, which is one of the properties of a parallelogram.

Or, there is always the longer way:

In rhombus $ABCD$ , all 4 sides are congruent (definition of a rhombus).

$AB\cong CD$ , $BC\cong DA$ , and $AC\cong AC$ .

By the SSS Postulate, $\triangle ABC\cong\triangle CDA$ .

Corresponding parts of congruent triangles are congruent, so $\angle BAC\cong BCA$ and $\angle B\cong\angle D$ , which is one of the properties of a parallelogram (opposite angles are congruent).

Proof that the diagonals of a rhombus divide it into 4 congruent triangles

In rhombus $ABCD$ , $M$ is the point at which the diagonals intersect.

Since the diagonals of a rhombus are bisectors of eachother, $AM\cong MC$ and $BM\cong MD$ .

Also, all sides are congruent.

By the SSS Postulate, the 4 triangles formed by the diagonals of a rhombus are congruent.

Proof that the diagonals of a rhombus are perpendicular

Continuation of above proof:

Corresponding parts of congruent triangles are congruent, so all 4 angles (the ones in the middle) are congruent.

This leads to the fact that they are all equal to $90^{\circ}$ degrees, and the diagonals are perpendiclar to eachother.

Example Problems

Introductory

2006 AMC 10B Problem 15

This article is a stub. Help us out by expanding it.

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 A '''rhombus''' is a geometric figure that lies in a [[plane]].  It is defined as a [[quadrilateral]] all of whose sides are [[congruent (geometry) | congruent]].  It is a special type of [[parallelogram]], and its properties (aside from those properties of parallelograms) include:
 * Its diagonals divide the figure into 4 congruent [[triangle]]s.
-* Its diagonals are [[perpendicular]].
+* Its diagonals are [[perpendicular]] [[bisect]]ors of eachother.
 * If all of a rhombus' [[angle]]s are [[right angle]]s, then the rhombus is a [[square (geometry) | square]].
@@ Line 7: / Line 7: @@
 ==Proofs==
-This article would be greatly enhanced by the proofs of the above facts.
+This article would be greatly enhanced by these proofs:
 ===Proof that a rhombus is a parallelogram===
+All sides of a rhombus are congruent, so opposite sides are congruent, which is one of the properties of a parallelogram.
+Or, there is always the longer way:
+In rhombus <math>ABCD</math>, all 4 sides are congruent (definition of a rhombus).
+<math>AB\cong CD</math>, <math>BC\cong DA</math>, and <math>AC\cong AC</math>.
+By the SSS Postulate, <math>\triangle ABC\cong\triangle CDA</math>.
+Corresponding parts of congruent triangles are congruent, so <math>\angle BAC\cong BCA</math> and <math>\angle B\cong\angle D</math>, which is one of the properties of a parallelogram (opposite angles are congruent).
 ===Proof that the diagonals of a rhombus divide it into 4 congruent triangles===
+In rhombus <math>ABCD</math>, <math>M</math> is the point at which the diagonals intersect.
+Since the diagonals of a rhombus are bisectors of eachother, <math>AM\cong MC</math> and <math>BM\cong MD</math>.
+Also, all sides are congruent.
+By the SSS Postulate, the 4 triangles formed by the diagonals of a rhombus are congruent.
 ===Proof that the diagonals of a rhombus are perpendicular===
+Continuation of above proof:
+Corresponding parts of congruent triangles are congruent, so all 4 angles (the ones in the middle) are congruent.
+This leads to the fact that they are all equal to <math>90^{\circ}</math> [[degree]]s, and the diagonals are perpendiclar to eachother.
 == Example Problems ==

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