The document is a student's report on circles and geometry that includes definitions of key circle terms like radius, diameter, chord, and arc. It also proves several circle theorems, such as equal chords subtending equal angles at the center and the perpendicular from the center bisecting a chord. Additionally, it discusses drawing a circle through three points and the relationship between arcs and their corresponding chords. The report aims to explain fundamental circle concepts and their properties.

1. CIRCLES
BY AAKANKSHA GUPTA
IX-D
10th grade

2. Table Of Contents
1. Circles And Its Related Terms.
2. Angle Subtended By A Chord At A Point.
3. Perpendicular From The Centre To A Chord.
4. Circle Through Three Points
5. Equal Chords And Their Distances From The Centre.
6. Angle Subtended By An Arc Of A Circle.
7. Cyclic Quadrilaterals.
8. What We Conclude?
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3. Introduction:
We come across many objects in our daily life which are round in
shape, such as 1 rupee coin, wheels of vehicles, bangles etc. How can
we define a circle? A circle is a closed figure in a plane and it is the
collection of all the points in the plane which are at a constant
distance from a fixed point in the plane. The fixed point is the center
of the circle and the constant distance is the radius of the circle.

4. Circles and its related terms
Circumference of a circle is the length of the complete circular curve
constituting the circle.
Chord of a circle is a line segment joining any two points on the
circle.
Arc of a circle is a part of the circle. Any two points A and B of a circle
divide the circle into two parts.
The smaller part is called the minor arc and the larger part is called
the major arc of the circle.
If the two parts are equal, AB is a diameter of the circle and each part
is called a semi circle.

5. Angle Subtended By A Chord At A Point:
We take a line segment PQ and a point R not on the
line containing PQ. Join PR and QR. Then ∠PRQ is
called the angle subtended by the line segment PQ at
the point R.ZPOQ is the angle subtended by the chord
PQ at the center O, ZPRQ and ZPSQ are respectively
the angles subtended by PQ at points R and S on the
major and minor arcs PQ

6. THEOREM 1
To Prove: Equal chords of a circle subtend equal angles at the center.
Proof: We are given two equal chords AB and CD of a circle with center O. We want to
prove that: Angle AOB = Angle COD.
In triangles AOB and COD,
OA = OC (Radii of a circle)
OB = OD (Radii of a circle)
AB = CD (Given)
Therefore, Triangle AOB = Triangle COD (SSS rule)
This gives ∠AOB = COD (CPCT)
;

7. If the angles subtended by the chords of a circle at
the center are equal, then the chords are equal.
Theorem 2 is converse of the Theorem 1.
THEORUM 2

8. Perpendicular From The Centre To A Chord:
p
o
Let there be a circle with O as center and let AB one of its chord.
Draw a perpendicular through O cutting AB at M .
Then , ZOMA LOMB = 90° or OM is perpendicular to AB.
We conclude that the perpendicular from the center of a
circle to a chord bisects the chord.

9. THEOREM 3
TO PROVE: The line drawn through the center of a circle to bisect a chord
is perpendicular to the chord.
PROOF: Let AB be a chord of a circle with center O and O is joined to the
mid-point M of AB.
We have to prove that OM perpendicular to AB.
Join OA and OB. In triangles OAM and OBM,
OA = OB (Given)
AM = BM (Given)
OM = OM (Common)
Therefore, ΔΟΑΜ = ΔΟΒΜ (SSS)
This gives angle OMA = angle LOMB = 90° (CPCT)

10. CIRCLE THROUGH THREE POINTS:
Infinite number of circle can be drawn through one or
two points (Fig 1 And 2).
In case there are three collinear points, then circle can
be drawn but the third point will lie outside the circle
(Fig 3).
Let us take three points A, B and C, which are not on the
same line (Fig 4).
Draw perpendicular bisectors of AB and BC, PQ and RS
respectively Let these perpendicular bisectors intersect
at one point O (Fig 5).

11. Figures
0
Fig.1
Fig.2
Fig.4
Fig.3
Fig.5

12. Equal Chords And Their Distances From The Centre:
Let AB be a line and P be a point. Since there are infinite
numbers of points on a line, if we join these points to P, we
will get infinitely many line segments PL1, PL2, PM, PL3, PL4,
etc.
Out of these line segments, the perpendicular from P to AB,
namely PM will be the least.
We conclude that the length of the perpendicular from a point
to a line is the distance of the line from the point P.

13. Theorum 4
TO PROVE: Equal chords of a circle (or of congruent circles) are equidistant from the center (or
centers).
PROOF: Take a circle of any radius. Draw two equal chords AB and CD of it and also the
perpendiculars OM and ON on them from the center O.
Divide the figure into two so that D falls on B and C falls on A . We observe that O lies on the
crease and N falls on M.
Therefore, OM = ON.

14. Theorem 5
Chords equidistant from the center
of a circle are equal in length.
Theorem 5 is converse of the
Theorem 4

15. Angle Subtended By An Arc Of A Circle:
The end points of a chord other than diameter of a circle cuts it into two arcs-one
major and other minor.
If we take two equal chords, They are more than just equal in length. They are
congruent in the sense that if one arc is put on the other, without bending or
twisting, one superimposes the other completely.
We can verify this fact by cutting the arc, corresponding to the chord CD from the
circle along CD and put it on the corresponding arc made by equal chord AB. We
will find that the arc CD superimpose the arc AB completely This shows that equal
chords make congruent arcs and conversely congruent arcs make equal chords of
a circle.
We conclude that if two chords of a circle are equal, then their corresponding arcs
are congruent and conversely, if two arcs are congruent, then their corresponding
chords are equal.

16. Theorem 6
TO PROVE: The angle subtended by an arc at the center is double the angle
subtended by it at any point on the remaining part of the circle.
PROOF: Given an arc PQ of a circle subtending angles POQ at the center O and
PAQ at a point A on the remaining part of the chord
Consider the three different cases arc PQ is minor (Fig 1), arc PQ is a
semicircle(Fig 2) and in arc PQ is major fig.3
Let us begin by joining AD and extending it to a point B all the cases,
angle BOQ= angle OAQ + angle AQO because an exterior angle of a triangle is
equal to the sum of the two interior opposite angles.
In ΔOAQ,
OQ=OA [Radii of a circle]
Therefore, angle OAQ = angleQOA
This gives, angle BOQ=2(angleOAQ)
Similarly, angle BOP =2 angle OAP (2)
From (1) and (2), angle BOP + angle BOQ=2( angle OAP+ angle OAQ)
This is the same as angle POQ =2 angle PAQ (3)
For the case (iii), where PQ is the major arc, (3) is replaced by reflex angle POQ
= 2 < PAQ.

17. Jupiter's rotation period
9h 55m 23s
Distance between Earth and the Moon
386,000 km
The Sun’s mass compared to Earth’s
333,000

18. Student’s geometry performance
Venus is beautiful and the second-
brightest natural object in the
night sky after the Moon
70%
Engaging in activities
Mercury is also the closest planet
to the Sun and the smallest in the
entire Solar System
50%
Score improvement

19. Our schools
School 1
Jupiter was named after
the god of the skies
School 3
Despite being red, Mars is
actually a cold place
School 2
Venus is the second
planet from the Sun

20. History of geometry
Euclidean geometry Despite being red, Mars is
actually a cold place
300 BCE
Non-euclidean Jupiter was named after the
Roman god of the skies
19th CE
Pythagorean theorem Earth is the third planet
from the Sun and has life
6th CE
Archimedes Venus has a beautiful
name, but it’s hot
3rd CE

21. Important geometry formulas
Despite being red, Mars is actually a
cold place. It’s full of iron oxide dust,
which gives the planet its reddish cast
c2=a2+b2
Pythagorean theorem
Earth is the third planet from the Sun
and the only one that harbors life in
the Solar System
A=½ x base x height
Area of a triangle

22. Properties of geometry shapes
Shape Definition Properties
Triangle A 3-sided polygon
⃞ We all live on Earth
⃞ Mercury is very small
Quadrilateral A 4-sided polygon
⃞ Venus is a hot planet
⃞ Mars is a cold planet
Circle
A round shape
with constant radius
⃞ Jupiter is a gas giant
⃞ Saturn has rings
Polygon
A closed, flat shape with
straight sides
⃞ Neptune is an ice giant
⃞ Earth is a blue planet

23. Follow the link in the graph to modify its data and then paste the new one here. For more info, click here
Ways to improve geometry comprehension
Practice
Venus is the second
planet from the Sun
61%
Visualize
Earth is also known as
the Blue Planet
57%
Examples
Despite being red, Mars
is actually a cold place
43%
Ask questions
Jupiter was named after
the god of the skies
22%

24. Our teachers
You can speak a bit about
this person here
Jenna Doe
You can speak a bit about
this person here
Susan Bones
You can speak a bit about
this person here
Timmy Jimmy

25. Constructing shapes
Use the lines provide to make a pentagon. You can rotate them if necessary
Lines My pentagon

26. Identify the shape
Nonagon
Match the words with the correct shape
Pentagon
Heptagon
Triangle

27. Area and perimeter quiz
Calculate the area and perimeter of the given shapes below. Type your answer in the text box
Area Write your answer here
Perimeter Write your answer here
Side A:
Side B:
Side C:
9 cm
7 cm
12 cm
Area Write your answer here
Perimeter Write your answer here
Length:
Width:
10 cm
6 cm

28. Problem solving methods
Saturn is also a gas giant
and the biggest planet in the
Solar System
Proportions
Mars is full of iron oxide
dust, giving the planet its
reddish cast
Ratios
Mercury is the closest planet
to the Sun and the smallest
in the System
Theorems
Venus is the second-
brightest natural object in
the night sky
Diagrams
Jupiter is a gas giant and the
biggest planet in the entire
Solar System
Similarity

29. Conclusions
This planet's name has nothing to do with the
liquid metal, since Mercury was named after
the Roman messenger god
⃞ Venus has a toxic atmosphere
⃞ Earth is the third planet from the Sun
⃞ Mars was named after a god

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