Class 9
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Mathematics
I’m Up and Down, and Round and Round
NCERT Solutions
NCERT Solutions
I’m Up and Down, and Round and Round
46 Solutions
Exercise:
All Exercises
End-of-Chapter Exercises
Exercise Set 5.1
Exercise Set 5.2
Exercise Set 5.3
Exercise Set 5.4
Exercise Set 5.5
Exercise Set 5.6
Think, Draw and Infer
Q1
End-of-Chapter Exercises
In a circle, a chord is 5 cm away from the centre. If the radius of the circle is 13 cm, what is the length of the chord?
Q2
End-of-Chapter Exercises
An arc of a circle subtends an angle of 70° at the centre. What is the measure of the angle subtended by the arc at a point on the circle?
Q3
End-of-Chapter Exercises
The diameter of a circle is 26 cm. A chord of length 24 cm is drawn in the circle. Find the distance from the centre of the circle to the chord.
Q4
End-of-Chapter Exercises
A circle has a radius of 15 cm. A chord is drawn. The distance from the centre of the circle to the chord is 9 cm. What is the length of the chord?
Q5
End-of-Chapter Exercises
Prove that the perpendicular bisector of a chord passes through the centre of the circle.
Q6
End-of-Chapter Exercises
The diameter of a circle is AB. Point C is on the circumference. What is the measure of the
∠
A
C
B
\angle \mathrm{ACB}
∠
ACB
? Explain your reasoning.
Q7
End-of-Chapter Exercises
ABCD is a cyclic quadrilateral inscribed in a circle. If
∠
A
\angle \mathrm{A}
∠
A
measures 75°, what is the measure of
∠
C
\angle \mathrm{C}
∠
C
? If
∠
B
\angle \mathrm{B}
∠
B
measures 110°, what is the measure of
∠
D
\angle \mathrm{D}
∠
D
?
Q8
End-of-Chapter Exercises
Quadrilateral PQRS is inscribed in a circle. If
∠
P
=
(
2
x
+
10
)
∘
\angle \mathrm{P}=(2 x+10)^{\circ}
∠
P
=
(
2
x
+
10
)
∘
and
∠
R
=
(
3
x
−
20
)
∘
\angle \mathrm{R}=(3 x-20)^{\circ}
∠
R
=
(
3
x
−
20
)
∘
, find the value of
x
x
x
and the measures of
∠
P
\angle \mathrm{P}
∠
P
and
∠
R
\angle \mathrm{R}
∠
R
.
Q9
End-of-Chapter Exercises
The distance of a chord of length 16 cm from the centre of a circle is 6 cm . Find the radius of the circle.
Q10
End-of-Chapter Exercises
A cyclic quadrilateral has sides 5, 5, 12, 12 units. Find its area.
Q11
End-of-Chapter Exercises
*11. Consider a cyclic quadrilateral. Without drawing its circumcircle, how can we find out whether the centre of the circumcircle lies inside the quadrilateral or outside? What is the best way of finding out?
Q12
End-of-Chapter Exercises
*12. When two chords intersect, each of them is divided into two line segments. Show that if the intersecting chords are of equal length, then the line segments of one chord are equal to the corresponding line segments of the other chord.
Q13
End-of-Chapter Exercises
*13. Draw a circle in which a chord of 6 cm length stands at a distance of 3 cm from the centre.
Q14
End-of-Chapter Exercises
*14. Show that rectangle is the only parallelogram that can be inscribed in a circle.
Q15
End-of-Chapter Exercises
*15. Show that if a rectangle is inscribed in a circle, then the point of intersection of its diagonals must lie at the centre of the circle.
Q16
End-of-Chapter Exercises
*16. Consider all chords of a circle of a fixed length. What is the shape formed by the midpoints of all these chords?
Q17
End-of-Chapter Exercises
*17. In a circle with centre O, chords AB and AC are congruent. Explain why this statement is true: "The centre of the circle lies on the angle bisector of
∠
B
A
C
\angle \mathrm{BAC}
∠
BAC
".
Q18
End-of-Chapter Exercises
Two parallel chords of lengths 10 cm and 24 cm are on the same side of the centre of a circle. The distance between the chords is 7 cm . Find the radius of the circle.
Q19
End-of-Chapter Exercises
*19. A regular hexagon is inscribed in a circle of radius
r
r
r
. Find the length of the sides of the hexagon and the distance of each side from the centre of the circle.
Q20
End-of-Chapter Exercises
A quadrilateral MNOP is inscribed in a circle. If MN is a diameter, what can you say about
∠
M
O
P
\angle \mathrm{MOP}
∠
MOP
and
∠
M
N
P
\angle \mathrm{MNP}
∠
MNP
? Explain your reasoning.
Q21
End-of-Chapter Exercises
Let ABCD be a cyclic quadrilateral. Explain why the exterior angle at any vertex is equal to the interior opposite angle (e.g.,
∠
C
D
E
=
∠
A
B
C
\angle \mathrm{CDE}=\angle \mathrm{ABC}
∠
CDE
=
∠
ABC
, where E is a point on the extension of side CD).
Q22
End-of-Chapter Exercises
*22. "There is no chord of a circle that is longer than its diameter." How do you justify this statement?
Q23
End-of-Chapter Exercises
*23. Let A be any point within a given circle with centre O . Show that the shortest chord of the circle that passes through point A is the one that is perpendicular to OA.
Q24
End-of-Chapter Exercises
How would you use the following figure to justify the statement that the angle in a semicircle is 90°?
Q25
End-of-Chapter Exercises
*25. In a circle, two chords CC' and DD' are drawn perpendicular to a diameter AB. Prove that the segment MM' joining the midpoints of the chords CD and C' D' is perpendicular to AB.
Q26
End-of-Chapter Exercises
*26. How would you use the following figure to justify the statement that the sum of the opposite angles of a cyclic quadrilateral is 180°?
Q1
Exercise Set 5.1
Draw
△
A
B
C
\triangle \mathrm{ABC}
△
ABC
with
A
B
=
5
c
m
,
∠
A
=
70
∘
\mathrm{AB}=5 \mathrm{~cm}, \angle \mathrm{~A}=70^{\circ}
AB
=
5
cm
,
∠
A
=
7
0
∘
and
∠
B
=
60
∘
\angle \mathrm{B}=60^{\circ}
∠
B
=
6
0
∘
. Draw the circumcircle of
△
A
B
C
\triangle \mathrm{ABC}
△
ABC
. Is the centre inside or outside the triangle?
Q2
Exercise Set 5.1
Draw
△
A
B
C
\triangle \mathrm{ABC}
△
ABC
with
A
B
=
5
c
m
,
∠
A
=
100
∘
,
A
C
=
4
c
m
\mathrm{AB}=5 \mathrm{~cm}, \angle \mathrm{A}=100^{\circ}, \mathrm{AC}=4 \mathrm{~cm}
AB
=
5
cm
,
∠
A
=
10
0
∘
,
AC
=
4
cm
. Draw the circumcircle of
△
A
B
C
\triangle \mathrm{ABC}
△
ABC
. Is the centre inside or outside the triangle?
Q3
Exercise Set 5.1
Draw
△
A
B
C
\triangle \mathrm{ABC}
△
ABC
, with
A
B
=
6
c
m
,
B
C
=
7
c
m
\mathrm{AB}=6 \mathrm{~cm}, \mathrm{BC}=7 \mathrm{~cm}
AB
=
6
cm
,
BC
=
7
cm
and
C
A
=
7
c
m
\mathrm{CA}=7 \mathrm{~cm}
CA
=
7
cm
. Draw the circumcircle of
△
A
B
C
\triangle \mathrm{ABC}
△
ABC
. Let the circumcentre be O. Measure OA, OB, OC.
Q4
Exercise Set 5.1
What is the least possible radius of a circle through two points A and B?
Q1
Exercise Set 5.2
Show that the triangle formed by a chord and the centre of the circle is isosceles.
Q2
Exercise Set 5.2
Show that if two such isosceles triangles (occurring in the previous question) have equal base length, they are congruent to each other.
Q1
Exercise Set 5.3
Can you explain why the converse to Theorem 4 is true, i.e., why does the perpendicular from the centre of a circle to a chord of the circle bisect the chord?
Q2
Exercise Set 5.3
An isosceles triangle ABC is inscribed in a circle, with AB = AC. Show that the altitude from A to BC passes through the centre of the circle.
Q3
Exercise Set 5.3
Two parallel chords of lengths 6 cm and 8 cm are on opposite sides of the centre of a circle. If the radius of the circle is 5 cm, find the distance between the midpoints of the chords.
Q1
Exercise Set 5.4
Use the Baudhāyana-Pythagoras theorem to show why Theorem 6 must be true.
Q2
Exercise Set 5.4
Consider a circle with centre C. If CE is perpendicular to chord AB, CH is perpendicular to chord GF, and CE = CH, show that AB = GF.
Q3
Exercise Set 5.4
Solve the previous question using the Baudhāyana-Pythagoras theorem.
Q1
Exercise Set 5.5
Find the length of the chord of a circle where the radius is 7 cm and perpendicular distance is 6 cm.
Q2
Exercise Set 5.5
Explain why the following statement is true: If the perpendicular distance of a chord from the centre is
d
d
d
and the radius is
r
r
r
, then the chord length is
2
r
2
−
d
2
2 \sqrt{r^{2}-d^{2}}
2
r
2
−
d
2
.
Q3
Exercise Set 5.5
In a circle, if the distance of chord AB from the centre is twice the distance of another chord CD from the centre, then can we conclude that
C
D
=
2
A
B
\mathrm{CD}=2 \mathrm{AB}
CD
=
2
AB
? Give reasons for your answer.
Q1
Exercise Set 5.6
In a circle with centre O, the central angle AOB is 60°. If the radius of the circle is 12 cm, what is the length of the chord AB?
Q2
Exercise Set 5.6
Let A and B be two points on a circle with centre O.
(i)
Are there points X, Y on the circle, on the same side of AB, such that
∠
A
X
B
\angle \mathrm{AXB}
∠
AXB
is different from
∠
A
Y
B
\angle \mathrm{AYB}
∠
AYB
?
(ii)
Is it true that if
∠
A
X
B
=
∠
A
Y
B
\angle \mathrm{AXB}=\angle \mathrm{AYB}
∠
AXB
=
∠
AYB
, then X and Y lie on the same side of the circle?
(iii)
If
∠
A
X
B
=
∠
A
Y
B
\angle \mathrm{AXB}=\angle \mathrm{AYB}
∠
AXB
=
∠
AYB
, and X and Y do not lie on the circle, does the circle through A, B and X also pass through Y?
Q3
Exercise Set 5.6
Find
x
x
x
in the given figure.
Q1
Think, Draw and Infer
A, B and C are three collinear points. Can you find a point P such that
P
A
=
P
B
=
P
C
\mathrm{PA}=\mathrm{PB}=\mathrm{PC}
PA
=
PB
=
PC
? What can you say about the perpendicular bisectors of AB and BC? Draw and check. Can you show that for three collinear points A, B and C, the perpendicular bisector of AB and BC are parallel? Is it possible for a circle to pass through collinear points? Can you draw a line that cuts a given circle in three distinct points?
Q2
Think, Draw and Infer
The circumcircle of a given
△
A
B
C
\triangle \mathrm{ABC}
△
ABC
is drawn. Can there be other triangles congruent to
△
A
B
C
\triangle \mathrm{ABC}
△
ABC
that share the same circumcircle?
More from this chapter
Chapter overview
Important Points
Practice Questions
Flashcards