Lesson PROPERTIES OF RHOMBIS

Source code of 'PROPERTIES OF RHOMBIS'

This Lesson (PROPERTIES OF RHOMBIS) was created by by ikleyn(52780)

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<H2>Properties of rhombis</H2>

By the definition, a rhombus is a parallelogram which has all the sides of the same length.

As a parallelogram, the rhombus has all the properties of a parallelogram:
&nbsp;&nbsp;&nbsp;&nbsp;- the opposite sides are parallel;
&nbsp;&nbsp;&nbsp;&nbsp;- the opposite sides are of equal length;
&nbsp;&nbsp;&nbsp;&nbsp;- each diagonal divides it in two congruent triangles;
&nbsp;&nbsp;&nbsp;&nbsp;- the diagonals bisect each other;
&nbsp;&nbsp;&nbsp;&nbsp;- the opposite angles are congruent;
&nbsp;&nbsp;&nbsp;&nbsp;- the sum of any two consecutive angles is equal to 180�.

Rhombis (plural of rhombus) have additional properties.

In a rhombus, each diagonal divides it in two congruent isosceles triangles.

In a rhombus, the two diagonals are perpendicular.

In a rhombus, the diagonals are the angle bisectors.

For your convenience, this file contains the annotations to my lessons on rhombis in this site and the major properties of rhombis. The properties 
are presented with the links to the corresponding lessons. The lessons are listed in the logical order, which means that every given lesson refers to the preceding ones 
and does not refer to that follow. The list consolidates the relevant lessons that are located under different topics in this site. 
At the end of the list there are links to the lessons on word problems related to rhombis.
<TABLE>
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 <TD>
The lesson title&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

<A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Diagonals-of-a-rhombus-are-perpendicular.lesson>Diagonals of a rhombus are perpendicular</A>

<A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Diagonals-of-a-rhombus-bisect-its-angles.lesson>Diagonals of a rhombus bisect its angles</A>

<A HREF=http://www.algebra.com/algebra/homework/word/geometry/The-length-of-diagonals-of-a-rhombus.lesson>The length of diagonals of a rhombus</A>

<A HREF=http://www.algebra.com/algebra/homework/word/geometry/A-circle-inscribed-to-the-rhombus.lesson>A circle inscribed in the rhombus</A>

......
 </TD>
 <TD>
The property&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

In a rhombus, the two diagonals are perpendicular.
If in a parallelogram the two diagonals are perpendicular, 
then the parallelogram is a rhombus.

In a rhombus, the diagonals are the angle bisectors.
If in a parallelogram the two diagonals are the angle bisectors, 
then the parallelogram is a rhombus.
If in a parallelogram the diagonal bisects an interior angle, 
then this diagonal bisects the opposite interior angle too, 
and the parallelogram is a rhombus.

If a is the length of the side of a rhombus 
and {{{d[1]}}} and {{{d[2]}}} are the lengths of its diagonals, then 
{{{d[1]^2 + d[2]^2}}} = {{{4a^2}}}.

If a is the length of the side of a rhombus 
and {{{d[1]}}} and {{{d[2]}}} are the lengths of its diagonals, then 
the radius {{{r}}} of the circle inscribed to the rhombus is equal to
{{{r = d[1]*d[2]/(4a)}}}.

......
 </TD>
 <TD>
Under the topic&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

Parallelograms

Geometry

......
 </TD>
 <TD>
In the section

Geometry

Word problems

......
 </TD>
 </TR>
</TABLE>

For the solutions of the typical word problems on rhombis related to their sides and diagonals measures see the lessons
&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/word/geometry/How-to-solve-problems-on-the-rhombus-sides-and-diagonals-measures-Examples.lesson>HOW TO solve problems on the rhombus sides and diagonals measures - Examples</A>,
&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/word/geometry/The-length-of-diagonals-of-a-rhombus.lesson>The length of diagonals of a rhombus</A> and 
&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/word/geometry/A-circle-inscribed-to-the-rhombus.lesson>A circle inscribed in the rhombus</A>
under the topic Geometry of the section Word problems in this site.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.