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.
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 , all 4 sides are congruent (definition of a rhombus).
, , and .
By the SSS Postulate, .
Corresponding parts of congruent triangles are congruent, so and . The same can be done for the two other angles, so .
Convert the congruences into measures to get and . Adding these two equations yields .
The interior angles of a quadrilateral add up to 360 degrees, so , or .
Substituting gives . When simplified, .
If two lines are cut by a transversal and same-side interior angles add up to 180 degrees, the lines are parallel. This means . The same can be done for the other two sides, and know we know that opposite sides are parallel. Therefore, a rhombus is a parallelogram.
Proof that the diagonals of a rhombus divide it into 4 congruent triangles
In rhombus , is the point at which the diagonals intersect.
Since the diagonals of a rhombus are bisectors of eachother, and .
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 degrees, and the diagonals are perpendicular to each other.