RS AGGARWAL CLASS 9 CHAPTER 6 INTRODUCTION TO EUCLID'S GEOMETRY EXERCISE 6

 EXERCISE 6

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Question 1:

What is the difference between a theorem and an axiom?

Answer 1:

An axiom is a basic fact that is taken for granted without proof.
Examples:
(i) Halves of equals are equal.
(ii) The whole is greater than each of its parts.

Theorem: A statement that requires proof is called theorem. 
Examples:
i) The sum of all the angles around a point is ${360}^{\circ }$.
ii) The sum of all the angles of triangle is ${180}^{\circ }$.        

Question 2:

Define the following terms:
(i) Line segment
(ii) Ray
(iii) Intersecting lines
(iv) Parallel lines
(v) Half line
(vi) Concurrent lines
(vii) Collinear points
(viii) Plane

Answer 2:

(i) Line segment :A line segment is a part of line that is bounded by two distinct end-points. A line segment has a fixed length.



(ii) Ray:  A line with a start point but no end point and without a definite length is a ray.



(iii) Intersecting lines: Two lines with a common point are called intersecting lines.



 (iv) Parallel lines: Two lines in a plane without a common point are parallel lines.



 (v) Half line: A straight line extending from a point indefinitely in one direction only is a half line.



 (vi) Concurrent lines: Three or more lines intersecting at the same point are said to be concurrent.

 (vii) Collinear points: Three or more than three points are said to be collinear if there is a line, which contains all the points.



 (viii) Plane: A plane is a surface such that every point of the line joining any two point on it, lies on it.

Question 3:

In the adjoining figure, name
(i) six points
(ii) five lines segments
(iii) four rays
(iv) four lines
(v) four collinear points

Answer 3:

(i) Points are A, B, C, D, P and R.

(ii) $\overline{EF}, \overline{GH}, \overline{FH} , \overline{EG}, \overline{MN}$

(iii) $\overrightarrow{EP}, \overrightarrow{GR}, \overrightarrow{HS,} \overrightarrow{FQ}$

(iv) $\overleftrightarrow{AB}, \overleftrightarrow{CD}, \overleftrightarrow{PQ}, \overleftrightarrow{RS}$

(v) Collinear points are M, E, G and B.

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Question 4:

In the adjoining figure, name:
(i) two pairs of intersecting lines and their corresponding points of intersection
(ii) three concurrent lines and their points of intersection
(iii) three rays
(iv) two line segments

Answer 4:

(i) Two pairs of intersecting lines and their point of intersection are
$\left\{\overleftrightarrow{EF}, \overleftrightarrow{GH}, point R\right\}, \left\{\overleftrightarrow{AB}, \overleftrightarrow{CD}, point P\right\}$

(ii) Three concurrent lines are
$\left\{\overleftrightarrow{AB}, \overleftrightarrow{EF}, \overleftrightarrow{GH}, point R\right\}$

(iii) Three rays are
$\left\{\overrightarrow{RB}, \overrightarrow{RH}, \overrightarrow{RF}\right\}$

(iv) Two line segments are
$\left\{\overline{RQ} and \overline{RP}\right\}$

Question 5:

From the given figure, name the following:
(a) Three lines
(b) One rectilinear figure
(c) Four concurrent points

Answer 5:

(a) $Line \overleftrightarrow{PQ}$, $Line\overleftrightarrow{RS}$ and $Line\overleftrightarrow{AB}$
(b) $CEFG$
(c) No point is concurrent.

Question 6:

(i) How many lines can be drawn through a given point?
(ii) How many lines can be drawn through two given points?
(iii) At how many points can two lines at the most intersect?
(iv) If A, B and C are three collinear points, name all the line segments determined by them.

Answer 6:

(i) Infinite lines can be drawn through a given point.


(ii) Only one line can be drawn through two given points.



(iii)  At most two lines can intersect at one point.




(iv) The line segments determined by three collinear points A, B and C are
$AB,\overline{BC}\text{and}AC.$

Question 7:

Which of the following statements are true?
(i) A line segment has no definite length.
(ii) A ray has no end-point.
(iii) A line has a definite length.
(iv) A line $\overleftrightarrow{AB}$ is same as line $\overleftrightarrow{BA}$.
(v) A ray $\underset{AB}{\to }$ is same as ray $\underset{BA}{\to }$.
(vi) Two distinct points always determine a unique line.
(vii) Three lines are concurrent if they have a common point.
(viii) Two distinct lines cannot have more than one point in common.
(ix) Two intersecting lines cannot be both parallel to the same line.
(x) Open half-line is the same thing as ray.
(xi) Two lines may intersect in two points.
(xii) Two lines are parallel only when they have no point in common.

Answer 7:

(i) False. A line segment has a definite length.

(ii) False. A ray has one end-point.

(iii) False. A line has no definite length.

(iv) True

(v) False. $\overleftrightarrow{BA}$ and $\overleftrightarrow{AB}$ have different end-points.
(vi) True

(vii) True

(viii) True

(ix) True

(x) True

(xi) False. Two lines intersect at only one point.

(xii) True

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Question 8:

In the given figure, L and M are the mid- points of AB and BC respectively.


(i) If AB = BC, prove that AL = MC.
(ii) If BL = BM, prove that AB = BC.

Hint
(i) $AB=BC\Rightarrow \frac{1}{2}AB=\frac{1}{2}BC\Rightarrow AL=MC$.
(ii) $BL=BM\Rightarrow 2BL=2BM\Rightarrow AB=BC$.

Answer 8:

(i) It is given that L is the mid-point of AB.            

∴ AL = BL = $\frac{1}{2}$AB       .....(1)

Also, M is the mid-point of BC.

∴ BM = MC = $\frac{1}{2}$BC     .....(2)

AB = BC       (Given)

⇒ $\frac{1}{2}$AB = $\frac{1}{2}$BC        (Things which are halves of the same thing are equal to one another)

⇒ AL = MC                [From (1) and (2)]


(ii) It is given that L is the mid-point of AB.            

∴ AL = BL = $\frac{1}{2}$AB

⇒ 2AL = 2BL = AB        .....(3)

Also, M is the mid-point of BC.

∴ BM = MC = $\frac{1}{2}$BC    

⇒ 2BM = 2MC = BC      .....(4)

BL = BM         (Given)

⇒ 2BL = 2BM         (Things which are double of the same thing are equal to one another) 

⇒ AB = BC             [From (3) and (4)]