**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.