Some Special Parallelograms

Rhombus

A rhombus is a quadrilateral with sides of equal length. Since the opposite sides of a rhombus have the same length, it is also a type of parallelogram.

An interesting property of a rhombus is that it is a special case of a kite and parallelogram. This is despite the fact that the properties of a kite and a parallelogram have nothing in common. Therefore, a rhombus consists of properties of a parallelogram and a kite as well.

As for the diagonals of a rhombus:

Property: The diagonals of a rhombus are perpendicular bisectors of one another.

Diagonals of a Rhombus

Instructions

Joining both the diagonals of the rhombus

• A rhombus is also a where we know, the diagonals each other.
• Therefore, considering the triangles- ∆AOB and ∆COB, we get OA = OCOB
• We also have AB = CBBO as they are of a rhombus and OB is a side.
• Thus, by congruency criterion- ∆ AOB ≅ ∆ COBOBC
• We also see that ∠AOB and ∠COB are equal and have an angle sum of °
• Thus, m ∠AOB = m ∠COB = °
• Therefore, the diagonals of a rhombus perpendicularly each other.

Example 7: RICE is a rhombus. Find x, y, z. Justify your findings.

Solution:

x = OE = OI (diagonals bisectdivide each other) =

y = OR = OC = (same reason as above)

z = side of the rhombus = (as all sides are equal)

Thus, x = 5 , y = 12 and z = 13

Rectangle

A rectangle is a type of parallelogram in which every angle equal. Therefore, it has all the properties of a parallelogram:

(1) Opposite sides of equal length and

(2) Diagonals bisect each other.

Since all the angles are equal in a rectangle and the total sum of the interior angles for a quadrilateral is 360°, we get that:

× Each angle = 360°

Each angle = 90°

Property: The diagonals of a rectangle are of equal length.

Diagonals of a Rectangle

Instructions

Joining both the diagonals of the rectangle

• Considering the triangles ∆ ABC and ∆BAD: AB = ABAD as it is a side.
• We also have: BC = AD as they are sides and m ∠A = m ∠B = ° as all angles are angles.
• Therefore, by criterion: ∆ ABC ≅ ∆ BADBDA
• Which gives us, AC = BDDB
• Therefore, in a rectangle the diagonals are equal in length and also bisect each other.

From the above conclusion, we can say that: In a rectangle the diagonals, besides being equal in length each other.

Example 8: RENT is a rectangle. Its diagonals meet at O. Find x, if OR = 2x + 4 and OT = 3x + 1

Solution:

OT is half of the diagonal TE and OR is half of the diagonal RN (as diagonals each other).

Diagonals are here. (RENT is a rectangleparallelogram)

So, their halves are also equal.

Therefore, 3x + 1 = 2x + 4 ⇒ x =

Square

A square is further, a special case of a , having equal sides.

Therefore, like earlier, a square has properties of a rectangle while having all sides equal.

In a square, the diagonals have the following properties:

(i) bisect one another (Square is a type of parallelogram)

(ii) are of equal length (Square is a type of rectangle) and

(iii) bisect each other perpendicularly.

So, now we need to prove the following property:

Property: The diagonals of a square are perpendicular bisectors of each other.

Diagonals of a square

Instructions

Joining both the diagonals of the square

• With O as the point of intersection for the diagonals, consider the triangles- ∆AOD and ∆COD
• We have: AD = CDOD as all sides are equal, OD is a side and OA = OCCD as diagonals each other.
• Therefore, by congruency criterion: ∆AOD ≅ ∆ CODDCO
• Thus we get, m∠AOD = m∠CODOCD
• Since the angles are a linear pair we get, m∠AOD = m∠COD = °
• Therefore, the diagonals of a square are bisectors of each other.

Let's Solve

A mason has made a concrete slab. He needs it to be rectangular. Answer Yes/No for the following below mentioned properties that need to be vertified in order to make sure that it is rectangular?

Instructions

Answer the following:

Instructions