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Quadrilaterals can be defined as, types of polygons that have four sides, four vertices, and four angles along with a pair of diagonals. The Sum of interior angles of quadrilaterals is 360°. There are various kinds of quadrilaterals. As the name itself suggest the word is a combination of two Latin words ‘Quadri‘ means a variant of four, and ‘latus‘ means side. In this topic, we are going to study some special types of parallelogram like rectangle, square, rhombus.
A parallelogram is a special type of quadrilateral whose opposite side are equal and parallel.
There are different properties of parallelograms.
Depending on the properties, there are three special types of parallelogram:
Rectangle
A rectangle is a special type of parallelogram which has all the properties of parallelogram along with some different properties. Each angle of a rectangle must be a right angle, i.e. of 90°.
Properties of a Rectangle
Properties of a rectangle are similar to those of a parallelogram:
Note: Rectangle has all the properties of parallelogram. So all rectangles are parallelogram but all parallelograms are not rectangle.
Problem 1: In rectangle ABCD, AO = 5cm. Find the length of diagonal BD. Also find the perimeter of rectangle if AB = 8cm and AD = 6cm.
Solution:
AO = OC = 5cm (diagonals bisect each other)
Therefore, AC = 10cm
BD = AC =10cm (diagonals of rectangle are equal)
Perimeter = AB + BC + CD + DA
= 8 + 6 +8 +6 (opposite sides are equal)
= 28cm
Problem 2: In rectangle ABCD, ∠ABD = 3x – 7 and ∠CBA = 6x – 2. Find the value of x.
Solution:
Each angle of rectangle is 90^{0}
Therefore,
∠ABD + ∠CBA = 90^{0}
3x – 7 + 6x – 2 = 90
9x – 9 = 90
9x = 99
x = 11
Problem 3: In rectangle ABCD AO = 2x – 10 cm, OB = x + 4 cm. Find the length of diagonal BD
Solution:
In rectangle diagonals bisect each other and are equal.
Therefore, AO = OB
2x – 10 = x + 4
x = 14
OB = 14 + 4 = 18 cm
OD = 18 cm (as diagonals bisect each other)
Therefore, BD = 36 cm
Rhombus
A quadrilateral that has all sides equal and opposite sides parallel is called Rhombus. Opposite angles of a rhombus are equal and diagonals of the Rhombus bisect each other perpendicularly. Note all rhombus are parallelograms but the reverse of this is not true.
Properties of a Rhombus
Properties of a rhombus are similar to those of a parallelogram:
Note: Rhombus is a parallelogram with all side equal
Problem 1: Diagonals of rhombus are 24cm and 10cm. Find the side of rhombus.
Solution:
AC = 24cm
BD = 10cm
Therefore, AO = 12cm and OB = 5cm (diagonals bisect each other)
In rightangled triangle AOB, (diagonals of rhombus are perpendicular)
AB^{2} = OA^{2} + OB^{2}
AB^{2} = 12^{2} + 5^{2}
AB^{2} = 144 + 25
AB^{2} = 169
AB = 13cm
Therefore, side of rhombus is 13cm.
Problem 2: In a rhombus one of the diagonals is equal to a side of the rhombus. Find the angles of rhombus.
Solution:
In rhombus PQRS PR = PQ (given)
Therefore, PQ = QR = RS = SP = PR (as all side of rhombus are equal)
In triangle PQR
PQ = QR = PR
Therefore, it is an equilateral triangle.
∠QPR = ∠Q = ∠QRP = 60^{0}
ly ∠SPR = ∠S = ∠PRS = 60^{0}
Therefore, angles of rhombus are ∠P = 120^{0 }, ∠Q = 60^{0}, ∠R = 120^{0}, ∠S = 60^{0}
Problem 3: Derive the formula for are of rhombus.
Solution:
As diagonals of rhombus bisect each other at right angle.
In rhombus ABCD
area of triangle ABD = 1/2 * BD *AO (1/2 * base *height) ………. (1)
area of triangle BCD = 1/2 * BD * CO ………………………………………….. (2)
Area of rhombus = Area of triangle ABD + area of triangle BCD
= 1/2 * BD * AO + 1/2 * BD * CO
= 1/2 * BD (AO + CO)
= 1/2 * BD * AC (AE + CE = AC)
Therefore, area of rhombus = 1/2 * product of diagonals
Square
A quadrilateral that has all sides equal and opposite sides parallel and all interior angles equal to 90° is called Square. Diagonals of a square bisect each other perpendicularly. Note that all squares are rhombus but not viceversa.
Properties of a Square
Properties of a square are similar to those of a rhombus:
Sample Problems on Square
Problem 1: All rhombus are squares or all squares are rhombus. Which of these statements is correct and why?
Solution:
Square and rhombus both have all sides equal but a rhombus is called square if each of its angle is 90^{0}. So all squares can be called rhombus but converse is not true.
Problem 2: In the figure ROPE is a square. Show that diagonals are equal.
Solution:
In Δ REP and Δ OEP
RE = OP (sides of square)
∠E = ∠P (each 90^{0})
EP = EP (common)
Therefore, triangles are congruent by SAS criteria.
Therefore, RP = OE (c.p.c.t)
Therefore, diagonals of square are equal.
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