7
The rhombus and its properties
Textbook: pp. 23–24
GoalKnow the definition and properties of a rhombus, including perpendicular diagonals.
New words
rhombus · rombperpendicular diagonals · perpendikulyar diagonallarbisector · bissektrisaside of a rhombus · romb tomoni
Explanation
A parallelogram with equal sides is called a rhombus. It has all the properties of a parallelogram and one more important property. The diagonals of a rhombus are perpendicular and they bisect the angles of the rhombus. In the proof we take the isosceles triangle BAD: AO is the median to its base, so it is also the bisector and the altitude. All four sides of a rhombus are equal, so its perimeter is P = 4a.
Worked examples
A diagonal of a rhombus makes a 28° angle with a side. The diagonal bisects the angle, so the angles of the rhombus are 56° and 180° − 56° = 124°.
The perimeter of a rhombus is 52 cm. All sides are equal, so a side is 52 : 4 = 13 cm.
Class activity
“Folded rhombus”: fold a sheet along a diagonal and notice that the diagonals are perpendicular and the angles are halved (use scissors only with an adult).
Practice
1
A side of a rhombus is 9 cm. What is the perimeter (cm)?
36
2
The perimeter of a rhombus is 60 cm. How long is a side (cm)?
15
3
In rhombus ABCD the diagonal AC makes a 40° angle with side AB. How many degrees is ∠A?
80
4
Why are the diagonals of a rhombus perpendicular?
Triangle BAD is isosceles (AB = AD) and AO is the median to its base, so it is also the altitude.