Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them?

Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them?

(i) parallel lines (ii) perpendicular lines (iii) line segment

(iv) radius of a circle (v) square

**Answer**

Yes, other terms need to be defined first which are:

Plane: A plane is flat surface on which geometric figures are drawn.

Point: A point is a dot drawn on a plane surface and is dimensionless.

Line: A line is collection of points which can extends in both direction and has only length not breadth.

(i) Parallel lines: When two or more never intersect each other in a plane and perpendicular distance between them is always constant then they are said to be parallel lines.

(ii) Perpendicular lines: When two lines intersect each other at right angle in a plane then they are said to be perpendicular to each other.

(iii) Line segment: A line segment is a part of a line with two end points and cannot be extended further.

(iv) Radius of circle: The fixed distance between the centre and the circumference of the circle is called the radius of the circle.

(v) Square: A square is a quadrilateral in which all the four sides are equal and each internal angle is a right angle.

(i) Given any two distinct points A and B, there exists a third point C which is in between A and B.

(ii) There exist at least three points that are not on the same line.

Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid’s postulates? Explain.

**Answer**

Undefined terms in the postulates:

→ Many points lie in a plane. But here it is not given about the position of the point C whether it lies on the line segment joining AB or not.

→ Also, there is no information about the plane whether the points are in same plane or not.

Yes, these postulates are consistent when we deal with these two situation:

(i) Point C is lying in between and on the line segment joining A and B.

(ii) Point C not lies on the line segment joining A and B.

No, they don't follow from Euclid’s postulates. They follow the axioms.

If a point C lies between two points A and B such that AC = BC, then prove that AC = 1/2 AB. Explain by drawing the figure.

Here, AC = BC

Now, adding AC both sides.

AC + AC = BC + AC

also, BC +AC = AB (as it coincides with line segment AB)

∴ 2 AC = AB (If equals are added to equals, the wholes are equal.)

⇒ AC = 1/2 AB.

In Question 4, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.

Let A and B be the line segment and points P and Q be two different mid points of AB.

Now,

∴ P and Q are midpoints of AB.

Therefore AP=PB and also AQ = QB.

also, PB + AP = AB (as it coincides with line segment AB)

Similarly, QB + AQ = AB.

Now,

AP + AP = PB + AP (If equals are added to equals, the wholes are equal.)

⇒ 2 AP = AB --- (i)

Similarly,

2 AQ = AB --- (ii)

From (i) and (ii)

2 AP = 2 AQ (Things which are equal to the same thing are equal to one another.)

⇒ AP = AQ (Things which are double of the same things are equal to one another.)

Thus, P and Q are the same points. This contradicts the fact that P and Q are two different mid points of AB. Thus, it is proved hat every line segment has one and only one mid-point.