# NCERT Solutions For Class 9 Math Chapter – 5 Exercise – 5.1

NCERT Solutions For Class 9 Math Chapter – 5 Exercise – 5.1

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Q1.

Which of the following statements are true and which are false ? Give reasons for your answers.

• Only one line can pass through a single point.
• There are an infinite number of lines which pass through two distinct points.
• A terminated line can be produced indefinitely on both sides.
• If two circles are equal , then their radii are equal.
• In Fig  If AB = PQ and PQ = XY then AB = XY.

Solution:-

• There can be infinite line drawn passing through  a single point.
•  Only one line can be drawn which passes through two distinct points.
• A terminated line can be produced indefinitely on both sides In geometry A line can be extended in both direction. A line means infinite long length.
• If two circles are equal , then their  radii are equal.

By suspension we will find that the centre ad circumference of the both circles coincide. Hence , their radius must be equal.

• By Euclid first axiom things which are equal to the same things are equal to one another.

Q2. 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 ?

• Parallel lines        (ii) Perpendicular lines       (iii) Line segment      (iv) Radius of a circle    (v) Square

Solution:-

Yes, other terms need to be a 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.

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

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

Line segment: A line segment is a part of a line with two ends points and cannot be extended further.

Radius of circle: The fixed distance between the center and the circumference o the circle is called the radius of circle.

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

Q3. Consider two ‘Postulates’ given below:

• Given any two distinct points A and B there exists third point C which is in between A and B.
• There exist at least three points that  are not on the same line.

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

Solution:-

Undefined terms in the postulates:

Many points lie in a plane But here 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:

• Point C is lying in between and on the line segment joining A and B.
• Point C not lies on the line segment joining A and B.

Q4. If a point C lies between two points points A and B such that AC = BC then prove that AC = 1/2 AB. Explain by drawing the figure.

Solution:-

Here AC = BC

AC + AC = BC + AC

Also , BC + AC = AB (As it coincides with line segment AB)

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

AC = 1/2 AB.

Q5. 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.

Solution:-

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 its coincides with line segment AB)

Similarly QB + AQ = AB.

Now ,

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

2AP – AB (i)

Similarly

2AQ = AB  (ii)

From (I) and (ii)

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

AP = PQ (Things which are doubled of the same things are equal to one another.)

Thus P and Q are the same point 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.

Q6. In Fig 5.10 If AC = BD then prove that AB = CD.

Solution:-

Given , AC = BD

From the figure,

AC = AB + BC

BD = BC + CD

AB + BC = BC + BD

According to Euclid axiom when equal are subtracted from equal , remainders are also equal.

Subtracting BC both sides,

AB + BC – BC = BC + CD – BC

AB = CD

Q7. Why is axiom 5 , in the list Euclid axioms , considered a universal truth?

Solution:-

Axiom 5: The whole is always greater than the part .

Take an example of a cake. When it is whole it will measure 2 pound but when we took out a part from it and measures its weight it will come out lower than the previous one. So , the fifth axiom of Euclid is true for all the universal things That is  why it considered a ‘universal truth’.