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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 :-
(a) All squares are rectangles but all rectangles can’t be squares, so this statement is false.

(b) All rhombuses are kites but all kites can’t be rhombus.

(c) True

(d) True

(e) True; squares fulfill all criteria of being a rectangle because all angles are right angle and opposite sides are equal. 
    Similarly, they fulfill all criteria of a rhombus, as all sides are equal and their diagonals bisect each other.

(f) False; All trapeziums are parallelograms, but all parallelograms can’t be trapezoid.

(g) False; all squares are parallelograms

(h) True
    

Question-2 :-  Identify all the quadrilaterals that have.
(a) four sides of equal length    (b) four right angles

Solution :-
(a) If all four sides are equal then it can be either a square or a rhombus.

(b) All four right angles make it either a rectangle or a square.
    

Question-3 :-  Explain how a square is.
(i) a quadrilateral   (ii) a parallelogram   (iii) a rhombus   (iv) a rectangle

Solution :-
(i) Having four sides makes it a quadrilateral

(ii) Opposite sides are parallel so it is a parallelogram

(iii) Diagonals bisect each other so it is a rhombus

(iv) Opposite sides are equal and angles are right angles so it is a rectangle.
    

Question-4 :-  Name the quadrilaterals whose diagonals.
(i) bisect each other
(ii) are perpendicular bisectors of each other
(iii) are equal

Solution :-
 Rhombus; because, in a square or rectangle diagonals don’t intersect at right angles.
    

Question-5 :-  Explain why a rectangle is a convex quadrilateral.

Solution :-
 A rectangle is a convex quadrilateral because Both diagonals lie in its interior, so it is a convex quadrilateral.
    

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). triangle

Solution :-
  If we extend BO to D, we get a rectangle ABCD. Now AC and BD are diagonals of the rectangle. In a rectangle diagonals are equal and bisect each other.
  So, AC = BD
  AO = OC
  BO = OD
  And AO = OC = BO = OD
  So, it is clear that O is equidistant from A, B and C.
    
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