Exercise 11A
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Question 1:
Name all the line segments in each of the following figures:
(i) Figure
(ii) Figure
(iii) Figure
ANSWER:
(i) The line segments are
YX ¯¯¯¯¯¯This is because it has two end points Y and X.YZ ¯¯¯¯¯¯ This is because it has two end points Y an Z.YX ¯This is because it has two end points Y and X.YZ ¯ This is because it has two end points Y an Z.
(ii)
AD¯¯¯¯¯This is because it has two end points A and D.AB¯¯¯¯¯ This is because it has two end points A and B.AC¯¯¯¯¯ This is because it has two end points A and C.AE¯¯¯¯¯ This is because it has two end points A and E.DB¯¯¯¯¯ This is because it has two end points B and D.BC¯¯¯¯¯ This is because it has two end points B and C.CE¯¯¯¯¯ This is because it has two end points C and E.AD¯This is because it has two end points A and D.AB¯ This is because it has two end points A and B.AC¯ This is because it has two end points A and C.AE¯ This is because it has two end points A and E.DB¯ This is because it has two end points B and D.BC¯ This is because it has two end points B and C.CE¯ This is because it has two end points C and E.
(iii)
PS¯¯¯¯¯ This is because it has two end points P and S.PQ¯¯¯¯¯ This is because it has two end points P and Q.QR¯¯¯¯¯ This is because it has two end points Q and R.RS¯¯¯¯¯ This is because it has two end points R and S.PR¯¯¯¯¯ This is because it has two end points P and R.QS¯¯¯¯¯ This is because it has two end points Q and S.PS¯ This is because it has two end points P and S.PQ¯ This is because it has two end points P and Q.QR¯ This is because it has two end points Q and R.RS¯ This is because it has two end points R and S.PR¯ This is because it has two end points P and R.QS¯ This is because it has two end points Q and S.
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Question 2:
Identify and name the line segments and rays in each of the following figures:
(i) Figure
(ii) Figure
(iii) Figure
ANSWER:
(i) Line segment is AB¯¯¯¯¯. This is because it has two end points A and B.AB¯. This is because it has two end points A and B.
Rays are:
−→AC This is because it has only one end point A.−→BD This is because it has only one end point B.→AC This is because it has only one end point A.→BD This is because it has only one end point B.
(ii) Line segments are:
EP¯¯¯¯¯ This is because it has two end points Eand P.EG¯¯¯¯¯ This is because it has two end points E and G.GP¯¯¯¯¯ This is because it has two end points G and P.EP¯ This is because it has two end points Eand P.EG¯ This is because it has two end points E and G.GP¯ This is because it has two end points G and P.
Rays are:
EF−→− This is because it has only one end point, i.e. E.GH−→− This is because it has only one end point, i.e. G.PQ−→− This is because it has only one end point, i.e. P.EF→ This is because it has only one end point, i.e. E.GH→ This is because it has only one end point, i.e. G.PQ→ This is because it has only one end point, i.e. P.
(iii) Line segments are:
OL¯¯¯¯¯ This is because it has two end points O and L.OP¯¯¯¯¯ This is because it has two end points O and P.OL¯ This is because it has two end points O and L.OP¯ This is because it has two end points O and P.
Rays are:
LM−→− This is because it has only one end point, i.e. L.PQ−→− This is because it has only one end point, i.e. P.LM→ This is because it has only one end point, i.e. L.PQ→ This is because it has only one end point, i.e. P.
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Question 3:
In the adjoining figure, name
(i) four line segments;
(ii) four rays;
(iii) two non-intersecting line segments.
Figure
ANSWER:
(i)
PR¯¯¯¯¯ This is because it has two end points P and R.QS¯¯¯¯¯ This is because it has two end points Q and S.PQ¯¯¯¯¯ This is because it has two end points P and Q.RS¯¯¯¯¯ This is because it has two end points R and S.PR¯ This is because it has two end points P and R.QS¯ This is because it has two end points Q and S.PQ¯ This is because it has two end points P and Q.RS¯ This is because it has two end points R and S.
(ii)
PA−→− This is because it has only one end point, i.e. P.RB−→− This is because it has only one end point, i.e. R.QC−→− This is because it has only one end point, i.e. Q.SD−→− This is because it has only one end point, i.e. S.PA→ This is because it has only one end point, i.e. P.RB→ This is because it has only one end point, i.e. R.QC→ This is because it has only one end point, i.e. Q.SD→ This is because it has only one end point, i.e. S.
(iii)
PR ¯¯¯¯¯¯and QS¯¯¯¯¯ are the two non−intersecting line segments as they do not have any point in common.PR ¯and QS¯ are the two non-intersecting line segments as they do not have any point in common.
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Question 4:
What do you mean by collinear points?
(i) How many lines can you draw passing through three collinear points?
(ii) Given three collinear points A, B, C. How many line segments do they determine? Name them.
Figure
ANSWER:
COLLINEAR POINTS :
Three or more points in a plane are said to be collinear if they all lie in the same line. This line is called the line of collinearity for the given points.
(i) We can draw only one line passing through three collinear points.
(ii) 3 Line segments are:
AB¯¯¯¯¯ This is because it has two end points A and B.BC¯¯¯¯¯ This is because it has two end points B and C.AC¯¯¯¯¯ This is because it has two end points A and C.AB¯ This is because it has two end points A and B.BC¯ This is because it has two end points B and C.AC¯ This is because it has two end points A and C.
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Question 5:
In the adjoining figure, name:
(i) four pairs of intersecting lines
(ii) four collinear points
(iii) three noncollinear points
(iv) three concurrent lines
(v) three lines whose point of intersection is P
Figure
ANSWER:
(i)
PS←→ and AB ←→− intersecting at S.CD←→and RS←→ intersecting at R.PS←→ and CD←→ intersecting at P.AB←→ and RS←→ intersecting at S.PS↔ and AB ↔ intersecting at S.CD↔and RS↔ intersecting at R.PS↔ and CD↔ intersecting at P.AB↔ and RS↔ intersecting at S.
(ii) A, Q, S and B are four collinear points as they all lie on the same line AB ←→−AB ↔.
(iii) A, C and B are non-collinear points as they do not lie on the same line.
(iv)
PS←→ , RS←→ and AB←→ are three concurrent lines passing through the same point SPS↔ , RS↔ and AB↔ are three concurrent lines passing through the same point S.
(v)
PS←→ , PQ←→ and CD←→ have common point of intersection PPS↔ , PQ↔ and CD↔ have common point of intersection P.
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Question 6:
Mark three noncollinear points A, B, C, as shown. Draw lines through these points taking two at a time. Name the lines. How many such different lines can be drawn?
Figure
ANSWER:
Taking points A and B, we can draw only one line AB ←→−AB ↔.
Taking points B and C, we can draw only one line BC←→ BC↔ .
Taking points A and C, we can draw only one line AC ←→−AC ↔.
We can draw only three lines through these non-collinear points A ,B and C.
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Question 7:
Count the number of line segments drawn in each of the following figures and name them.
(i) Figure
(ii) Figure
(iii) Figure
(iv) Figure
ANSWER:
(i) There are 6 line segments. These are:
AB¯¯¯¯¯ (with end points A and B)AC¯¯¯¯¯ (with end points A and C)AD¯¯¯¯¯ (with end points A and D)BC¯¯¯¯¯ (with end points B and C)BD¯¯¯¯¯ (with end points B and D)CD¯¯¯¯¯ (with end points C and D)AB¯ (with end points A and B)AC¯ (with end points A and C)AD¯ (with end points A and D)BC¯ (with end points B and C)BD¯ (with end points B and D)CD¯ (with end points C and D)
(ii) There are 10 line segments. These are:
AB¯¯¯¯¯ (with end points A and B)BC ¯¯¯¯¯¯ (with end points B and C)CD¯¯¯¯¯ (with end points C and D)AD¯¯¯¯¯ (with end points A and D)AC¯¯¯¯¯ (with end points A anc C)BD ¯¯¯¯¯¯ (with end points B and D)AO ¯¯¯¯¯¯ (with end points A and O)CO¯¯¯¯¯ (with end points C and O)BO¯¯¯¯¯ (with end points B and O)DO¯¯¯¯¯ (with end points D and O)AB¯ (with end points A and B)BC ¯ (with end points B and C)CD¯ (with end points C and D)AD¯ (with end points A and D)AC¯ (with end points A anc C)BD ¯ (with end points B and D)AO ¯ (with end points A and O)CO¯ (with end points C and O)BO¯ (with end points B and O)DO¯ (with end points D and O)
(iii) There are 6 line segments. They are:
AB¯¯¯¯¯, AF¯¯¯¯¯, FB¯¯¯¯¯, EC¯¯¯¯¯, ED¯¯¯¯¯, DC¯¯¯¯¯AB¯, AF¯, FB¯, EC¯, ED¯, DC¯
(iv) There are 12 line segments. They are:
AB¯¯¯¯¯, AD¯¯¯¯¯, AE¯¯¯¯¯BC¯¯¯¯¯, BF¯¯¯¯¯ CG¯¯¯¯¯, CD¯¯¯¯¯HG¯¯¯¯¯¯, HE¯¯¯¯¯¯ , DH¯¯¯¯¯¯EF¯¯¯¯¯, GF¯¯¯¯¯AB¯, AD¯, AE¯BC¯, BF¯ CG¯, CD¯HG¯, HE¯ , DH¯EF¯, GF¯
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Question 8:
Consider the line PQ←→PQ↔ given below and find whether the given statements are true or false:
(i) M Is a point on ray NQ−→−NQ→.
(ii) L is a point on ray MP−→−MP→.
(iii) Ray MQ−→−MQ→ is different from ray NQ−→−NQ→.
(iv) L, M, N are points on line segment LN¯¯¯¯¯LN¯.
(v) Ray LP−→LP→ is different from ray LQ−→−LQ→.
Figure
ANSWER:
(i) False
M is outside ray NQ.
(ii) True
L is placed between M and P.
(iii) True
Ray MQ is extended endlessly from M to Q and ray NQ is extended endlessly from N to Q.
(iv) True
(v) True
LP −→−is extended endlessly from L to P.LQ−→− is extended endlessly from L to Q.LP →is extended endlessly from L to P.LQ→ is extended endlessly from L to Q.
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Question 9:
Write ‘T’ for true and ‘F’ for false in case of each of the following statements:
(i) Every point has a size.
(ii) A line segment has no length.
(iii) Every ray has a finite length.
(iv) The ray AB−→−AB→ is the same as the ray BA−→−BA→.
(v) The line segment AB¯¯¯¯¯AB¯ is the same as the line segment BA¯¯¯¯¯BA¯.
(vi) The line AB←→AB↔ is the same as the line BA←→BA↔.
(vii) Two points A and B in a plane determine a unique line segment.
(viii) Two intersecting lines intersect at a point.
(ix) Two intersecting planes intersect at a point.
(x) If points A, B, C are collinear and points C, D, E are collineaer then the pints A,B, C, D, E are collinear.
(xi) One and only one ray can be drawn with a given end point.
(xii) One and only one line can be drawn to pass through two given points.
(xiii) An unlimited number of lines can be drawn to pass through a given point.
ANSWER:
(i) False
A point does not have any length, breadth or thickness.
(ii) False
A line segment has a definite length.
(iii) False
A ray has no definite length.
(iv) False
Ray AB has initial point A and is extended endlessly towards B, while ray BA has initial point B and is extended endlessly towards A.
(v) True
This is because both the line segments have definite length with end points A and B.
(vi) True
This is because it neither has a definite length nor any end point.
(vii) True
Only one line segment can pass through the two given points.
(viii) True
(ix) False
Two intersecting planes intersect at a line.
(x) False
Different set of collinear points need not be collinear.
(xi) False
With point P, endless rays (like PA, PB, PC, PD, PE, PF) can be drawn.
(xii) True
Two points define one unique line.
(xiii) True
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Question 10:
Fill in the blanks:
(i) A line segment has a ………….. length.
(ii) A ray has ………….. end point.
(iii) A line has ………….. end point.
(iv) A ray has no ………….. length.
(v) A line ………….. be drawn on a paper.
ANSWER:
(i) definite
(ii) one
(iii) no
(iv) definite
(v) cannot
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Exercise 11B
Question 1:
Which of the following has no end points?
(a) A line segment
(b) A ray
(c) A line
(d) None of these
ANSWER:
(c) A line does not have any end point. It is a line segment that is extended endlessly on both sides.
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Question 2:
Which of the following has one end point?
(a) A line
(b) A ray
(c) A line segment
(d) None of these
ANSWER:
(b) A ray has one end point, which is called the initial point. It is extended endlessly towards the other direction.
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Question 3:
Which of the following has two end points?
(a) A line segment
(b) A ray
(c) A line
(d) None of these
ANSWER:
(a) A line segment has two end points and a definite length that can be measured.
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Question 4:
Which of the following has definite length?
(a) A line
(b) A line segment
(c) A ray
(d) None of these
ANSWER:
(b) A line segment has a definite length that can be measured by a ruler and, therefore, it can be drawn on a paper.
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Question 5:
Which of the following can be drawn on a piece of paper?
(a) A line
(b) A line segment
(c) A ray
(d) A plane
ANSWER:
(b) A line segment has a definite length that can be measured by a ruler. So, it can be drawn on a paper.
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Question 6:
How many lines can be drawn passing through a given point?
(a) One only
(b) Two
(c) Three
(d) Unlimited number
ANSWER:
(d) Unlimited number of lines can be drawn.
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Question 7:
How many lines can be drawn passing through two given point?
(a) One only
(b) Two
(c) Three
(d) Unlimited number
ANSWER:
(a) Only one line can be drawn that passes through two given points.
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Question 8:
Two planes intersect
(a) at a point
(b) in a plane
(c) in a line
(d) none of these
ANSWER:
(c) Two intersecting planes intersect in a line.
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Question 9:
Two lines intersect
(a) at a point
(b) at two points
(c) at an infinite number of points
(d) in a line
ANSWER:
(a) Two lines intersect at a point.
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Question 10:
Two points in a plane determine
(a) exactly one line segment
(b) exactly two line segments
(c) an infinite number of line segments
(d) none of these
ANSWER:
(a) exactly one line segment
Two points in a plane determine exactly one line segment with those two points as its end points.
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Question 11:
The minimum number of points of intersection of three lines in a plane is
(a) 1
(b) 2
(c) 3
(d) 0
ANSWER:
(d) 0
Three lines will not necessarily intersect in a plane. Thus, the minimum point of intersection will be 0.
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Question 12:
The maximum number of points of intersection of three lines in a plane is
(a) 0
(b) 1
(c) 2
(d) 3
ANSWER:
(d) 3
The maximum number of points of intersection of three lines that intersect in a plane are three.
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Question 13:
Choose the correct statement:
(a) every line has a definite length
(b) every ray has a definite length
(c) every line segment has a definite length
(d) none of these
ANSWER:
(c) Every line segment has a definite length.
Every line segment has a definite length, which can be measured using a ruler.
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Question 14:
Choose the false statement:
(a) Line AB←→AB↔ is the same as line BA←→BA↔
(b) Ray AB−→−AB→ is the same as ray BA−→−BA→
(c) Line segment AB¯¯¯¯¯AB¯ is the same as teh line segment BA¯¯¯¯¯BA¯
(d) None of these
ANSWER:
(b) Ray AB−→− is same as ray BA−→− AB→ is same as ray BA→
This is because the initial points in these rays are A and B, respectively, and are extended endlessly towards B and A, respectively.
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Question 15:
How many rays can be drawn with a given point as the initial point?
(a) One
(b) Two
(c) An unlimited number
(d) A limited number only
ANSWER:
(c) An unlimited number of rays can be drawn with a given point as the initial point. For example:
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