Excercise 3.4 Understanding Quadrilaterals NCERT Solutions Class 8
Understanding Quadrilaterals
Exercise 3.4
Question 1
State whether True or False.
(a) All rectangles are squares.
(b) All rhombuses are parallelograms.
(c) All squares are rhombuses and also rectangles.
(d) All squares are not parallelograms.
(e) All kites are rhombuses.
(f) All rhombuses are kites.
(g)All parallelograms are trapeziums.
(h) All squares are trapeziums.
Solution
Steps:

Shapes 
True or False 
Reason 
A 
All rectangles are Squares. 
False 
A rectangle need not have all sides equal hence it is not square. 
B 
All rhombuses are parallelograms 
True 
Since the opposite sides of a rhombus have the same length, it is also a parallelogram 
C 
All squares are rhombuses and also rectangles. 
True 
All squares are rhombuses as all sides of a square are of equal lengths. A square is a rectangle as each internal angle is \(90\) degrees. 
D 
All squares are not parallelograms. 
False 
The opposite sides of a parallelogram are of equal length hence squares with all sides equal are paralleolgrams. 
E 
All kites are Rhombuses. 
False 
Since rhombus have all sides of equal length. All kites need not have all sides of the same length. 
F 
All rhombuses are kites. 
True 
Since, all rhombuses have equal sides and diagonals bisect each other. 
G 
All parallelograms are trapeziums 
True 
Since, trapezium has only two parallel sides. 
H 
All squares are Trapeziums. 
True 
All trapezium have a pair of parallel sides hence squares can be trapezium. 
Question 2
Identify all the quadrilaterals that have.
(a) four sides of equal length
(b) four right angles
Solution
Steps:
a) Four sides of equal length  Rhombus and Square are the quadrilaterals with \(4\) sides of equal length.
b) Four right angles  Square and Rectangle are the quadrilaterals with \(4\) right angles.
Question 3
Explain how a square is.
(i) a quadrilateral
(ii) a parallelogram
(iii) a rhombus
(iv) a rectangle
Solution
Steps:
(i) 
Quadrilateral 
A square is a quadrilateral since it has four sides. 
(ii) 
Parallelogram properties (1) Opposite sides are equal. (2) Opposite angles are equal. (3) Diagonals bisect one another. 
A square is parallelogram, since it contains both pairs of opposite sides equal. 
(iii) 
Rhombus  properties i)A parallelogram with sides of equal length. ii)The diagonals of a rhombus are perpendicular bisectors of one another. 
A square is a rhombus since i) its four sides are of same length. ii)the diagonals of a square are perpendicular bisectors of each other. 
(iv) 
Rectangleproperties i)Being a parallelogram, the rectangle has opposite sides of equal length and its diagonals bisect each other. 
A square is rectangle since each interior angle measures \(90\) degree. 
Question 4
Name the quadrilaterals whose diagonals.
(i) bisect each other
(ii) are perpendicular bisectors of each other
(iii) are equal
Solution
Steps:
(i) bisect each other
Parallelogram:
Rhombus:
Rectangle:
Square:
a) Parallelogram
b) Rhombus
c) Rectangle
d) Square
The diagonals of a parallelogram, rhombus, rectangle and square are perpendicular bisectors of each other.
ii) Are perpendicular bisectors of each other
a) Rhombus
b) Square
The diagonals of a square and rhombus are perpendicular bisectors of each other.
(iii) are equal
a) Rectangle
b) Square
The diagonals of a rectangle and square are equal.
Question 5
Explain why a rectangle is a convex quadrilateral.
Solution
Steps:
Polygons that are convex have no portions of their diagonals in their exteriors. A rectangle is a convex quadrilateral since its vertex are raised and both of its diagonals lie in its interior.
Or
None of the angles being a reflex angle, So rectangle is convex quadrilateral.
Question 6
\(ABC\) is a rightangled triangle and \(O\) is the midpoint of the side opposite to the right angle. Explain why \(O\) is equidistant from \(A, B\) and \(C\). (The dotted lines are drawn additionally to help you).
Solution
What is Known?
\(ABC\) is a rightangled triangle and \(O\) is the midpoint of the side opposite to the right angle.
What is Unknown?
Why \(O\) is equidistant from \(A, B\) and \(C\)
Reasoning:
Since, two right triangles make a rectangle and in any rectangle, diagonals bisect each other.
Steps:
\(ABCD\) is a rectangle as opposite sides are equal and parallel to each other and all the interior angles are of \(90^\circ .\)
\[\begin{align}{\rm{AD}}\left {\left {{\rm{BC}},{\rm{AB}}} \right} \right{\rm{DC}}\\{\rm{AD }} = {\rm{BC}},{\rm{ AB }} = {\rm{ DC}}\end{align}\]
In a rectangle, diagonals are of equal length and also these bisect each other.
Hence, \({\rm{AO }} = {\rm{ OC }} = {\rm{ BO }} = {\rm{ OD}}\)
Since, two right triangles make a rectangle where \(O\) is equidistant point from \(A, B, C\) and \(D\) because \(O\) is the midpoint of the two diagonals of a rectangle.
So, \(O\) is equidistant from \(A, B, C \) and \(D.\)
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