Exercise
5.1
1. What
is the disadvantage in comparing line segments by mere observation?
Solution
When we compare two line segments
by observation, the minute difference between the two of them cannot be
observed and hence there are more chances of errors.
2. Why
is it better to use a divider than a ruler, while measuring the length of a
line segment?
Solution
A ruler and a divider can be used
to measure the length of a line segment. But, it is better to use a divider for
an accurate measure. The thickness of the ruler and the positioning of the eye
may cause difficulties in reading the correct measure.
3. Draw
any line segment, say AB. Take any point C lying in between A and B. Measure
the lengths of AB, BC and AC. Is AB = AC + CB? [Note : If A,B,C are any three
points on a line such that AC + CB = AB, then we can be sure that C lies
between A and B.]
Solution
Consider a line segment AB of
length = 8 cm.
Now, mark a point C at AC = 5 cm
Measure CA.
We see that the length of CA = 3 cm
i.e., AB = AC + CB
Therefore, for any point C lying in
between A and B on the line AB, the above is true.
4. If
A, B, C are three points on a line such that AB = 5 cm, BC = 3 cm and AC = 8
cm, which one of them lies between the other two?
Solution
AB = 5 cm
BC = 3 cm
AC = 8 cm
From the given measures, we can see
that
AB + BC = 8 cm = AC
So, B lies in between A and C.
5. Verify, whether D
is the mid-point of line segment AG.
Solution
From the figure above,
AD = 4 – 1 = 3 cm
DG = 7 – 4 = 3 cm
Since, AD = DG = 3 cm, D is the
mid-point of the line segment AG.
6.
If B is the
midpoint of AC and C is the
midpoint of BD, where A, B, C, D
lie on straight line, say why AB = CD?
Solution
B is the midpoint of AC. (Given)
So, AB = BC (1)
C is the midpoint of BD. (Given)
So, BC = CD (2)
From (1) and (2), AB = CD.
Exercise 5.2
1.
What fraction of a
clockwise revolution does the hour hand of a clock turn through, when it goes
from
a)
3 to 9 (b) 4 to 7
(c) 7 to 10(d) 12 to 9 (e) 1 to 10 (f) 6 to 3
Solution
When the hand of a
clock moves from one position to another, it turns through an angle. When it
moves from 12 to 12, it moves four right angles, that is, 360° which is one
revolution.
When it moves from
12 to 6, it moves two right angles, that is, 180°. This is 180/360 = ½ of one
revolution.
a)
3 to 9
When the hour hand
goes from 3 to 9, it moves or rotates by two right angles, that is 180°.
Fraction of
revolution = 180°/360° = ½
b)
4 to 7
When the hour hand
of the clock goes from 4 to 7, it rotates by 1 right angle, that is 90°.
Fraction of
revolution = 90°/360° = ¼
c)
7 to 10
When the hour hand
of the clock goes from 7 to 10, it rotates by 1 right angle, that is 90°.
Fraction of
revolution = 90°/360° = ¼
d)
12 to 9
When the hour hand
of the clock goes from 12 to 9, it rotates by 3 right angles, that is 270°.
Fraction of
revolution = 270°/360° = ¾
e)
1 to 10
When the hour hand
of the clock goes from 1 to 10, it rotates by 3 right angles, that is 270°.
Fraction of
revolution = 270°/360° = ¾
f)
6 to 3
When the hour hand
of the clock goes from 6 to 3, it rotates by 3 right angles, that is 270°.
Fraction of revolution
= 270°/360° = ¾
2.
Where will the hand
of a clock stop if it
(a) starts at 12
and makes ½ of a revolution, clockwise?
(b) starts at 2 and
makes ½ of a revolution, clockwise?
(c) starts at 5 and
makes ¼ of a revolution, clockwise?
(d) starts at 5 and
makes ¾ of a revolution, clockwise?
Solution
A complete
revolution is measured as 360°.
½ of a revolution
is 360°/2 = 180°
¼ of a revolution
is 90°
¾ of a revolution
is 270°
a)
The hand of the
clock starts at 12 and makes ½ of a revolution. That is, it starts at 12 and
revolves 180° and stops at 6.
b)
The hand starts at
2 and makes ½ of a revolution. That is, it starts at 2 and revolves 180° and stops
at 8.
c)
When the hand starts
at 5 and makes ¼ of a revolution, that is, 90°, it stops at 9.
d)
When the hand starts
at 5 and makes ¾ of a revolution, that is 270°, it stops at 2.
3.
Which direction
will you face if you start facing
(a)
East and make ½ of
a revolution clockwise?
(b)
East and make 1 ½
of a revolution clockwise?
(c)
West and make ¾ of
a revolution anti-clockwise?
(d)
South and make one
full revolution?
(Should we specify clockwise or anti-clockwise for
this last question? Why not?)
Solution
a)
When you start facing east and make ½ of
a revolution, you revolve by 180°. Hence, you would be facing west.
b)
When you start facing east and make 1
revolution clockwise, you face east again. Then you make another ½ of a
revolution, that is, 180°. Hence, you would be facing west.
c)
When you start facing west and make ¾ of
a revolution anti-clockwise, you revolve by 270°. This is by 180° and then by
90°. When you revolve by 180° anti-clockwise you face east and then you revolve
by 90° anti-clockwise, you face north.
d)
When you start facing south and make a
full revolution, that is 360°, you will again face the south. The direction,
whether clockwise or anticlockwise, does not matter in this case because a full
revolution will bring back to the initial position.
4.
What part of a
revolution have you turned through if you stand facing
(a) east and turn
clockwise to face north?
(b) south and turn
clockwise to face east?
(c) west and turn
clockwise to face east?
Solution
The following image
shows the directions, north (N), south (S), east (E) and west (W). When you
stand facing north and turn clockwise to face east, you have turned by a right
angle, i.e. 90°.
So, you have made
1/4th of a complete revolution.
a)
You stand facing
east and turn clockwise to face north.
So, you have turned
90° + 90° + 90° = 270° from East to North in the clock-wise direction. This is
270°/360° = ¾ th of the complete revolution.
b)
You start from south
and turn clockwise to face east.
So, you have turned 90° + 90° + 90° = 270° from South to
East in the clock-wise direction. This is 270°/360° = ¾ th of the complete
revolution.
c)
You stand facing west
and turn clockwise to face east.
So, you have turned
90° + 90° = 180° from West to East in the clock-wise direction. This is
180°/360° = ½ of the complete revolution.
5.
Find the number of
right angles turned through by the hour hand of a clock when it goes from
(a)
3 to 6 (b) 2 to 8
(c) 5 to 11 (d) 10 to 1 (e) 12 to 9 (f) 12 to 6
Solution
When the hour hand
of a clock moves from 12 to 3, from 3 to 6, from 6 to 9 and from 9 to 12, it
makes a right angle. So, when the hour hand of a clock turns four right angles,
it is said to make a complete revolution, 360°.
a)
3 to 6
The hour hand turned 1 right angle.
b)
2 to 8
The hour hand turned two right angles. [from 2 to 5 one
right angle and 5 to 8 another right angle]
c)
5 to 11
The hour hand turned two right angles. [from 5 to 8 one
right angle and 8 to 11 another right angle]
d)
10 to 1
The hour hand turned one right angle.
e)
12 to 9
The hour hand turned 3 right angles. [From 12 to 3, 3 to
6 and from 6 to 9]
f)
12 to 6
The hour hand turned two right angles. [from 12 to 3 and
from 3 to 6]
6.
How many right
angles do you make if you start facing
(a) south and turn
clockwise to west?
(b) north and turn
anti-clockwise to east?
(c) west and turn
to west?
(d) south and turn
to north?
Solution
The following image
shows the directions, north (N), south (S), east (E) and west (W). When you
stand facing north and turn clockwise to face east, you have turned by a right
angle, i.e. 90°.
a)
When you start
facing south and turn clockwise to west you make one right angle.
b)
When you start
facing north and turn anti-clockwise to east, you make three right angles.
c)
You start facing
west and turn to west you have made one complete revolution, which is four
right angles.
d)
When you start
facing south and turn to north you have made two right angles.
7.
Where will the hour
hand of a clock stop if it starts
(a) from 6 and
turns through 1 right angle?
(b) from 8 and
turns through 2 right angles?
(c) from 10 and
turns through 3 right angles?
(d) from 7 and
turns through 2 straight angles?
Solution
a) When the hour hand
starts from 6 and turns through 1 right angle, it will stop at 9.
b) When the hour hand
starts from 8 and turns through 2 right angles, it will stop at 2. [8 to 11 is
one right angle, 11 to 2 is the second right angle]
c) When the hour hand
starts from 10 and turns through 3 right angles, it stops at 7. [10 to 1 is one
right angle, 1 to 4 is the second and 4 to 7 is the third right angle.]
d) When the hour hand starts from 7 and turns
through 2 straight angles, that is 360°, it stops at 7.
Exercise 5.3
1.
Match the
following:
i)
Straight angle
|
a)
Less than one-fourth of a revolution
|
ii)
Right angle
|
b)
More than half a revolution
|
iii)
Acute angle
|
c)
Half of a revolution
|
iv)
Obtuse angle
|
d)
One-fourth of a revolution
|
v)
Reflex angle
|
e)
Between ¼ and ½ of a revolution
|
f)
One complete revolution
|
Solution
Straight angle is 180° which is ½ of a complete
revolution. So, i) ↔ c)
Right angle is 90° which is ¼ of a complete revolution.
So, ii) ↔ d)
Acute angles are angles less than 90°. This is less than
one-fourth of a revolution. So, iii) ↔ a)
Obtuse angles are angles greater than 90° but less than
180°. This is between ¼ and ½ of a complete revolution. So, iv) ↔ e)
Reflex angles are angles greater than 180° but less than
360°, that is, more than ½ of a complete revolution. So, v) ↔ b)
Solution
The measure of a
right angle is 90° and hence that of a straight angle is 180°. An angle is acute
if its measure is smaller than that of a right angle and is obtuse if
its measure is greater than that of a right angle and less than a straight
angle. A reflex angle is larger than a straight angle.
Solution
a) Acute angle
b) Obtuse angle
c) Right angle
d) Reflex angle
e) Straight angle
f) Acute angle
Exercise 5.4
1.
What is the measure
of (i) a right angle? (ii) a straight angle?
Solution
a)
A right angle
measures 90°
b)
A straight angle
measures 180°
2.
Say True or False:
(a)
The measure of an
acute angle < 90°.
True
(b)
The measure of an
obtuse angle < 90°.
False, an obtuse angle is greater than 90°
(c)
The measure of a
reflex angle > 180°.
True
(d)
The measure of one
complete revolution = 360°.
True
(e)
If m<A = 53° and
m<B = 35°, then m<A > m<B.
True
3.
Write down the
measures of (a) some acute angles. (b) some obtuse angles. (give at least two
examples of each).
Solution
a)
Acute angles are
angles less than 90°. Some acute angles are 35°, 43°
b)
Obtuse angles are
angles greater than 90° and less than 180°. Some obtuse angles are 135°, 143°
6.
Fill in the blanks
with acute, obtuse, right or straight :
a)
An angle whose
measure is less than that of a right angle is ______.
acute angle
b) An angle whose
measure is greater than that of a right angle is ______.
obtuse angle
c) An angle whose
measure is the sum of the measures of two right angles is _____.
straight angle
d)
When the sum of the
measures of two angles is that of a right angle, then each one of them is
______.
45°
e) When the sum of the measures of
two angles is that of a straight angle and if one of them is acute then the
other should be _______.
obtuse
9.
Find the angle
measure between the hands of the clock in each figure
Solution
The angle measure of the clock showing 9.00 a.m. is 90°.
The angle measure of the clock showing 1.00 p.m. is 30°.
The angle measure of the clock showing 6.00 p.m. is 180°.
10. In the given
figure, the angle measures 30°. Look at the same figure through a magnifying
glass. Does the angle become larger? Does the size of the angle change?
Solution
When we look through the magnifying glass, the angle does
not become larger. It remains the same. The size of the angle does not change.
Exercise 5.5
1.
Which of the
following are models for perpendicular lines:
(a)
The adjacent edges
of a table top.
(b)
The lines of a railway
track.
(c)
The line segments
forming the letter ‘L’.
(d)
The letter V.
Solution
The adjacent edges
of a table top and the line segments forming the letter L are models of a
perpendicular line. (a) and (c) are perpendicular lines.
The lines of a railway track are models of parallel lines.
The letter V has lines that are inclined at an angle.
2. Let PQ be the perpendicular to the line segment XY. Let PQ and XY intersect in the point A. What is the measure of <PAY?
Solution
PQ is perpendicular to XY and it intersects at point A.
So, <PAY = 90°.
3. There are two
set-squares in your box. What are the measures of the angles that are formed at their corners? Do they have any angle measure that is common?
Solution
One set-square
measures 90°, 60°, 30°
Other set-square
measures 90°, 45°, 45°
90° is a common
measure.
4. Study the diagram.
The line l is perpendicular to line m
(a) Is CE = EG?
(b)
Does PE bisect CG?
(c)
Identify any two
line segments for which PE is the perpendicular bisector.
(d)
Are these true?
i.
AC > FG
ii.
CD = GH
iii.
BC < EH.
Solution
a)
CE = 2 units and EG
= 2 units. So, CE = EG.
b)
Yes, as CE = EG, we
say PE bisects CG.
c)
DF and BH are line
segments for which PE is the perpendicular bisector.
d)
i) True, as AC
is 2 units but FG is 1 unit.
ii) True, CD = GH
= 1 unit.
iii) True, BC is 1
unit and EH is 3 units.
Exercise 5.6
1. Name the types
of following triangles:
(a) Triangle with
lengths of sides 7 cm, 8 cm and 9 cm.
(b) ΔABC with AB =
8.7 cm, AC = 7 cm and BC = 6 cm.
(c) ΔPQR such that
PQ = QR = PR = 5 cm.
(d) ΔDEF with m<D
= 90°
(e) ΔXYZ with
m<Y = 90° and XY = YZ.
(f) ΔLMN with
m<L = 30°, m<M = 70° and m<N = 80°.
Solution
a)
Scalene triangle,
as the triangle has three unequal sides.
b)
Scalene triangle,
as the triangle has three unequal sides.
c)
Equilateral
triangle, as the triangle has three equal sides.
d)
Right-angled
triangle as one angle is a right angle.
e)
Right-angled
isosceles triangle as one angle is right angle and two sides are equal.
f)
Acute-angled
triangle as each angle is less than a right angle.
2. Match the
following
Measures of triangle
|
Type of triangle
|
i)
3 sides of equal length
|
(a)
Scalene triangle
|
ii)
2 sides of equal length
|
(b)
Isosceles right angled
|
iii)
All sides of different length
|
(c)
Obtuse angled
|
iv)
3 acute angles
|
(d)
Right angled
|
v)
1 right angle
|
(e)
Equilateral
|
vi)
1 obtuse angle
|
(f)
Acute angled
|
vii)
1 right angle with 2 sides of equal length
|
(g)
Isosceles
|
Solution
i) 3 sides of equal length – Equilateral
triangle (e)
ii) 2 sides of equal length – Isosceles (g)
iii) All sides of different length – Scalene
(a)
iv) 3 acute angles – Acute angled (f)
v) 1 right angle – Right angled (d)
vi) 1 obtuse angle – Obtuse angled (c)
viii)
1 right angle with
2 sides of equal length – Isosceles right angled (b)
3. Name each of the
following triangles in two different ways: (you may judge the
nature of the angle by observation)
Solution
a) Isosceles triangle, as two sides are of equal length.
Acute angled triangle, as angle measures are less than
right angle.
b)
Scalene triangle,
as three sides are of unequal length.
Right angled triangle, as one angle measures 90 degrees.
c)
Isosceles triangle,
as two sides are of equal length.
Obtuse angled triangle, as one angle is obtuse.
d)
Isosceles triangle,
as two sides are of equal length.
Right angled triangle, as one angle measures 90 degrees.
e)
Equilateral
triangle, as three sides are of equal length.
Acute angled triangle, as angle measures are less than
right angle.
f)
Scalene triangle,
as three sides are of unequal length.
Obtuse angled triangle, as one angle is obtuse.
5. Try to construct
triangles using match sticks. Some are shown here.
Can you make a triangle with
(a) 3 matchsticks?
(b) 4 matchsticks?
(c) 5 matchsticks?
(d) 6 matchsticks?
(Remember you have
to use all the available matchsticks in each case)
Name the type of
triangle in each case. If you cannot make a triangle, think of reasons for it.
Solution
A triangle can be
formed with 3, 5 and with 6 matchsticks.
A triangle cannot
be formed with 4 matchsticks as the sum of any two sides of a triangle has to
be greater than the third side. [A triangle can be formed with 2 sticks on one
side, 1 on the other and 1 on the third side. But, 1 + 1 = 2 which is not
greater than the third side and hence a triangle cannot be formed.]
Exercise 5.7
1. Say True or
False:
(a) Each angle of a
rectangle is a right angle.
(b) The opposite
sides of a rectangle are equal in length.
(c) The diagonals
of a square are perpendicular to one another.
(d) All the sides
of a rhombus are of equal length.
(e) All the sides
of a parallelogram are of equal length.
(f) The opposite
sides of a trapezium are parallel.
Solution
a)
True
b)
True
c)
True
d)
True
e)
False
f)
False
2. Give reasons for
the following:
(a) A square can be
thought of as a special rectangle.
(b) A rectangle can
be thought of as a special parallelogram.
(c) A square can be
thought of as a special rhombus.
(d) Squares,
rectangles, parallelograms are all quadrilaterals.
(e) Square is also
a parallelogram.
Solution
a) When a rectangle is
drawn with all equal sides, it becomes a square. Hence it can be thought of as
a special rectangle.
b) When a
parallelogram is drawn with all the angles as a right angle, it becomes a
rectangle. Hence, a rectangle is a special parallelogram.
c)
A rhombus and a
square have equal sides. When a rhombus is drawn with all the angles as a right
angle, it becomes a square.
d)
Squares, rectangle,
parallelograms are all closed shapes made of 4 line segments and are all quadrilaterals.
e)
A parallelogram has
two opposite sides that are parallel and equal. Similarly, a square has all
sides that are equal and the opposite sides are parallel. Hence, square can be
thought of as a special parallelogram.
3. A figure is said
to be regular if its sides are equal in length and angles are equal in measure.
Can you identify the regular quadrilateral?
Solution
A square is a
regular quadrilateral as all its sides are equal in length and its angles are
equal to a right angle.
Exercise 5.8
1. Examine whether
the following are polygons. If any one among them is not, say why?
Solution
a)
Is not a polygon as
it is not a closed figure.
b)
Is a polygon as it
is closed and is made of only line segments.
c)
Is not a polygon as
it is not made of line segments.
d)
Is not a polygon as
it is not made of line segments.
2. Name each
polygon. Make two more examples of each of these.
Solution
a)
Quadrilateral, as
it made of four line segments.
b)
Triangle, as it
made of three line segments.
c)
Pentagon, as it
made of five line segments.
d)
Octagon, as it made
of eight line segments.
3. Draw a rough
sketch of a regular hexagon. Connecting any three of its vertices, draw a
triangle. Identify the type of the triangle you have drawn.
The triangle formed
by connecting the vertices is an isosceles triangle.
4. Draw a rough
sketch of a regular octagon. (Use squared paper if you wish). Draw a rectangle
by joining exactly four of the vertices of the octagon.
5. A diagonal is a
line segment that joins any two vertices of the polygon and is not a side of
the polygon. Draw a rough sketch of a pentagon and draw its diagonals.
Exercise 5.9
1.
Match the following:
Solution
a)
Cone ii)
b)
Sphere iv)
c)
Cylinder v)
d)
Cuboid iii)
e)
Pyramid i)
2. What shape is
(a) Your instrument
box?
(b) A brick?
(c) A match box?
(d) A road-roller?
(e) A sweet laddu?
Solution
a)
Cuboid
b)
Cuboid
c)
Cuboid
d)
Cylinder
e)
Sphere
ex 5.2 -the third answer is 8 but marked as 9
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