Class 8 Maths NCERT Solutions for Chapter – 3 Understanding Quadrilaterals Ex – 3.4

Understanding Quadrilaterals

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) False
(b) True
(c) True
(d) False
(e) False
(f) True
(g) True
(h) True

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

Solution.
(a) The quadrilaterals having four sides of equal length is either a square or a rhombus. ,
(b) The quadrilaterals having four right angles is either a square or a rectangle.

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

Solution.
(i) A square is 4 sided, so it is a quadrilateral.
(ii) A square has its opposite sides parallel, so it is a parallelogram.
(iii) A square is a parallelogram with all the four sides equal, so it is a rhombus.
(iv) A square is a parallelogram with each angle a right angle, 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.
(i) The quadrilaterals whose diagonals bisect each other can be a parallelogram or a rhombus or a square or a rectangle.
(ii) The quadrilaterals whose diagonals are perpendicular bisectors of each other can be a rhombus or a square.
(iii) The quadrilaterals whose diagonals are equal can be a square or a rectangle.

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

Solution.
Since the measure of each angle is less than 180° and also both the diagonals of a rectangle he wholly in its interior, so a rectangle 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).

Solution.

Produce BO to D such that BO = OD. Join AD and DC. Then ABCD is a rectangle. In the rectangle ABCD, its diagonals AC and BD are equal and bisect each other at O.
∴ OA = OC and OB = OD.
But AC = BD
Therefore, OA = OB = OD
Thus, O is equidistant from A, B and C.