NCERT Class 8 Maths Chapter 3 Understanding Quadrilaterals Exercise 3.4

Chapter 3 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.
Answer:
a) False, as all square are rectangles but all rectangles are not square.
b) True
c) True
d) False, as all squares are parallelograms as opposite sides are parallel and opposite angles are equal.
e) False, as all sides of kite are not same.
f) True
g) True
h) True

Question 2: Identify all the quadrilaterals that have.
a) four sides of equal length
b) four right angles
Answer:
a) Rhombus and square
b) Square and rectangle

Question 3: Explain how a square is
i) a quadrilateral
ii) a parallelogram
iii) a rhombus
iv) a rectangle
Answer:
i) Square has four sides, so it is a quadrilateral.
ii) Square has opposite sides parallel, so it is a parallelogram.
iii) Square is a parallelogram with all 4 sides equal, so it is a rhombus.
iv) A square is a parallelogram with all 4 angles 90°, so it is a rhombus.

Question 4: Name the quadrilaterals whose diagonals.
i) bisect each other
ii) are perpendicular bisectors of each other
iii) are equal
Answer:
i) Parallelogram, Rhombus, Square and Rectangle
ii) Rhombus and Square
iii)Rectangle and Square

Question 5: Explain why a rectangle is a convex quadrilateral.
Answer: Convex quadrilaterals are the quadrilaterals that have their diagonals on interior of quadrilaterals. Rectangle has its diagonals in interior and hence it is a convex quadrilaterals.

Question 6: ABC is a right-angled triangle and O is the mid-point 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).

Answer:
OA = OC (given)

OB = OD (construction)
Join AD, DC

ΔAOB
ΔDOC

OA = OC (S)
OB = OD (S)

∠AOB = ∠DOC (Vertically Opposite Angles)
ΔAOB ≅ ΔDOC (SAS rule)

AB = DC (corresponding parts of congruent triangles)

AD = BC (ΔAOB ≅ ΔBOC) (corresponding parts of congruent triangles)

Therefore, ABCD is a rectangle. Diagonals are equal and bisect each other. Hence, O is equidistant from A, B, C and D.