Quadrilaterals

Exercise 3.4

Question 1: State whether True or False.

(a) All rectangles are squares

Answer: All squares are rectangles but all rectangles can’t be squares, so this statement is false.

(b) All kites are rhombuses.

Answer: All rhombuses are kites but all kites can’t be rhombus.

Answer: True

(d) All rhombuses are kites.

Answer: True

(e) All squares are rhombuses and also rectangles

Answer: 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) All parallelograms are trapeziums.

Answer: False; All trapeziums are parallelograms, but all parallelograms can’t be trapezoid.

(g) All squares are not parallelograms.

Answer: False; all squares are parallelograms

(h) All squares are trapeziums.

Answer: True

Question 2: Identify all the quadrilaterals that have.

(a) four sides of equal length (b) four right angles

Answer: (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

Answer: (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

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

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

Answer: 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.

Answer: 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.