Rhombus
A rhombus is a special quadrilateral characterized by having all four sides of equal length. As a special case of a kite, a rhombus not only shares properties with kites but is also classified as a parallelogram due to its equal opposite sides. This section discusses various properties of rhombuses, particularly focussing on the pivotal property that the diagonals of a rhombus are perpendicular bisectors of each other.
Key Properties of a Rhombus:
- Equal Sides: All sides are of the same length.
- Opposite Angles: Like all parallelograms, a rhombus has equal opposite angles.
- Diagonals: The diagonals of a rhombus bisect each other at right angles; they are also equal in length.
These properties not only define the rhombus's structure but also serve as foundational elements in solving various geometric problems. The significance of understanding a rhombus lies in its applications in different fields such as architecture, design, and various mathematical concepts.
Example 8:
Consider a rhombus ABCD (Fig. 3.31). Given that \( m \angle AOB = 60^\circ \), find the lengths of x, y, z, and justify your findings.
Solution:
\( x = AC \)
\( y = BD \)
\( z = \) side of the rhombus
\( OA = OB = OC = OD = 10 \) (all sides are equal)
\[ x = 10 \]
\[ y = 10 \]
\[ z = 10 \]
Example 9:
In a rhombus PQRS (Fig. 3.32), if \( m \angle PQR = 120^\circ \), calculate the values of a, b, and c, and justify your conclusions.
Solution:
\( a = PQ \)
\( b = QR \)
\( c = \) side of the rhombus
\( PQ = QR = PS = RS = 15 \)
\[ a = 15 \]
\[ b = 15 \]
\[ c = 15 \]
Example 10:
Examine the rhombus LMNO (Fig. 3.33) where \( m \angle LMO = 45^\circ \). Determine the lengths d, e, and f, providing justification for your results.
Solution:
\( d = LM \)
\( e = NO \)
\( f = \) side of the rhombus
\( LM = NO = LO = MN = 8 \)
\[ d = 8 \]
\[ e = 8 \]
\[ f = 8 \]