Well-designed AP Board Solutions Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry Exercise 5.1 offers step-by-step explanations to help students understand problem-solving strategies.

## Introduction to Euclid’s Geometry Class 9 Exercise 5.1 Solutions – 9th Class Maths 5.1 Exercise Solutions

Question 1.

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

(i) Only one line can pass through a single point.

(ii) There are an infinite number of lines which pass through two distinct points.

(iii) A terminated line can be produced indefinitely on both the sides.

(iv) If two circles are equal, then their radii are equal.

(v) In below figure, if AB = PQ and PQ = XY, then AB = XY.

Answer:

(i) False. We can draw infinite lines that passing through a given point as shown.

(ii) False, from the Euclidean Axiom, it is clear that there is a unique line that passes through the given two points.

(iii) True, from the Euclidean postulate (2) it is clear.

(iv) True, let us consider two equal circles. Then put one circle over the other. (Which means super impose the boundary of one circle over the other), then we observe their radii are equal. Since by super imposing two equal circles, we notice, centres, boundaries and radii coincides.

(v) True, it is clear from Euclids axiom (i).

“Things (AB, XY) which are equal to the same thing ’(PQ) are equal to one another”.

Hence AB = XY.

Question 2.

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:

(i) Parallel lines: Parallel lines are lines that always maintain the same distance apart and never intersect each other.

Parallel lines are two or more straight lines that lie on the same plane and never intersect each other, regardless how for they are-extended

(or)

Two or more lines are said to be parallel lines if they do not have a common point on them:

To define parallel lines, we need to define “distance” first.

Distance is a measurement of how ’ far apart of two lines in space. (That is shortest)

(ii) Perpendicular lines : Two lines are said to be perpendicular, if they intersect each other at a right angle.

l, m, n two lines, ‘O’ is intersecting point angle at ‘O’ is 90°.

Hence l, m are perpendicular lines.

(iii) Line segment: Part of a straight line is called line segment.

‘l’ is a line and \(\overline{\mathrm{AB}}\) is a part of line T.

So, we call AB is a line segment and is denoted by \(\overline{\mathrm{AB}}\).

\(\overline{\mathrm{AB}}\) will have a finite length

(iv) Radius of a circle : The distance between a point on boundary of a circle and its centre is called radius of a circle.

Let A, B, C, D, E are different points on the boundary of above circle and

‘O’ is the centre, then \(\overline{\mathrm{OA}}, \overline{\mathrm{OB}}, \overline{\mathrm{OC}}, \overline{\mathrm{OD}}, \overline{\mathrm{OE}}\) are radii of above circle. It is clear all above are equal.

\(\overline{\mathrm{OA}}=\overline{\mathrm{OB}}=\overline{\mathrm{OC}}=\overline{\mathrm{OD}}=\overline{\mathrm{OE}}\)

(v) Square : A quadrilateral having four equal sides and four equal interior angles.

\(\overline{\mathrm{AB}}=\overline{\mathrm{BC}}=\overline{\mathrm{CD}}=\overline{\mathrm{DA}}\)

∠A = ∠B = ∠C = ∠D

Here “quadrilateral” is needed to define first.

Quadrilateral : A plane- figure that has sides and four vertices.

Question 3.

Consider two ‘postulates’ given below:

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

(ii) There exists 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 are

(i) points, line etc. The position of ‘C’ is not given. Hence undefined.

(ii) Yes, they are consistent, because s there is no controversy between the given two postulates. These two postulates deal two different situations.

Do they follow Euclid’s postulates ?

- No, they are not derived from Euclid’s postulates.

But they follow Euclid’s following axiom. - There is a unique line passes through given two points.

Question 4.

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

Answer:

Given that ‘C lies between ‘A’ and ‘B’ such that AC = BC.

Let ‘C’ be a point on AB as shown in figure below.

and also AC = BC (given)

We can write AC + CB = AB

⇒ AC + AC = AB (∵ Given BC = AC)

⇒ 2AC = AB

∴ AC = \(\frac{AB}{2}\)

Hence AC = \(\frac{1}{2}\) AB

So, ‘C’ is mid point of AB.

Question 5.

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.

Answer:

Given that ‘C’ is mid point of AB.

So, AC = CB = \(\frac{1}{2}\) AB ………… (1)

Let ‘D’ be another mid point of AB.

Then AD = DB = \(\frac{1}{2}\) AB …………. (2)

Comparing (1) and (2) we observe “D” is in the place of C’.

(∵ AC = CB, AD = DB = \(\frac{1}{2}\) AB)

∴ ‘C’ and D’ should be same.

∴ There exist only one ,mid point to every segment.

So, mid point is unique to every line segment.

Question 6.

In adjacent figure, if AC = BD, then prove that AB = CD.

Answer:

Given figure

Given that AC = BD ……….. (1)

We can write AC = AB + BC and BD = BC + CD

Now putting values of AC, BD in (1).

We get AC = BD

AB + BC = BC + CD

⇒ AB = CD

Hence proved.

Question 7.

Why is axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate.)

Answer:

Euclid’s Axiom – 5 states that,

“The whole is greater than its part”. Since it holds true in any branch of mathematics and other sciences as well. Hence axiom 5 is considered a ‘Universal truth’.