A parallelogram whose four sides are all equal is called a rhombus.
Properties of Rhombuses
- All four sides are equal.
- The diagonals intersect at right angles (AC⊥BD).
- The diagonals bisect the respective angles.
Example ①
In rhombus ABCD, when AB=5cm and diagonal AC=8cm, find BO (O is the intersection of the diagonals).
Solution: BO² = AB² − AO² = 25 − 16 = 9. BO = 3 cm (when using the Pythagorean theorem)
In rhombus ABCD, when AB=5cm and diagonal AC=8cm, find BO (O is the intersection of the diagonals).
Solution: BO² = AB² − AO² = 25 − 16 = 9. BO = 3 cm (when using the Pythagorean theorem)
Key Point
- Rhombus ⇒ Properties of parallelograms + perpendicular diagonals
- Four equal sides ⇔ rhombus
Practice Problem
- Prove using congruence that the diagonals of a rhombus intersect at right angles.
Answer and Explanation
- Answer: In rhombus ABCD, let O be the intersection of the diagonals. In △AOB and △AOD / AB=AD (rhombus) / OB=OD (property of parallelograms) / AO=AO (common side) / △AOB≅△AOD by SSS. Therefore, ∠AOB=∠AOD, and since their sum is 180°, ∠AOB=90°.