If you happen to be viewing the article What are Circle Theorems?? on the website Math Hello Kitty, there are a couple of convenient ways for you to navigate through the content. You have the option to simply scroll down and leisurely read each section at your own pace. Alternatively, if you’re in a rush or looking for specific information, you can swiftly click on the table of contents provided. This will instantly direct you to the exact section that contains the information you need most urgently.
The circle theorem helps understand the concepts of different elements of the circle, like sectors, tangents, angles, chords, and radius of the ring with proofs. A circle is the joining line of all the points that lie at an equal distance from a fixed focus point. This fixed point is in the middle point inside the circle.
However, all the points on the circle are at equal distance, and hence this fixed point is known as the center of the circle. The length between the circle center point and any point that lies on the circle is known as the radius. The space occupied by the circle is its area, and the outer line of the circle is its circumference. The line that is perpendicular to the circle at any point on the circle is known as a tangent.
Contents
All Theorems Related to Circle
Now, let’s look into the circle theorems and circle theorem proof to define the relationships between different entities of the circle. Before getting into the theorems, let us discuss the chord as it will give a better understanding.
Know Chord of a Circle
(Image will be Uploaded Soon)
A chord is the line segment that connects two different points of the circle’s circumference. Also, the diameter is the most significant chord that transverses the center of the circle.
Now, let us study different theorems of circle class 9 related to the circle.
Circles Class 9 Theorems
Students from Class 9 come across the circle basics, and they will learn various theorems related to the circle that helps to study the chord of the circle. Below are the topics that include in different circle class 9 theorems:
-
Angle made by the chord of the circle at a point
-
The line segment is perpendicular to the chord to the center.
-
The distance of different chords from the circle’s center and equal chords
-
The angle created by the arch of the circle
-
Cyclic quadrilaterals
All Theorems of Circle Class 9
Theorem 1:
Chords, having equidistant from the circle’s center make equal angles at the circle’s center.
(Image will be Uploaded Soon)
Proof:
Consider ∆AOB and ∆POQ,
AB = PQ (Chords that are equal) ……..(Equation 1)
OA = OB = OQ = OP (Radius of the circle) …..(Equation 2)
From equation 1 and equation 2, we can conclude;
∆AOB ≅ ∆POQ (Axiom of congruence SSS)
Hence, by Corresponding parts of the congruent triangles (CPCT), we will get;
∠AOB = ∠POQ
Hence, Proved.
Theorem 1 Converse Rule:
When two angles sectioned at the circle’s center that are made by two different chords are equal, those two chords are the same in length.
(Image will be Uploaded Soon)
Proof:
Consider ∆AOB and ∆POQ,
∠AOB = ∠POQ (Angles are equal given in the theorem statement) …………( Equation 1)
OA = OB = OP = OQ (Radius of the same circle)…………(Equation 2)
From equation 1 and equation 2, we can conclude;
∆AOB ≅ ∆POQ (Axiom of congruence SAS)
Hence,
AB = PQ (By CPCT rule)
Theorem 2: Circle Geometry
When you draw the perpendicular line segment from the circle’s center, it will bisect the chord, i.e., the perpendicular will divide the chord into two equal parts. It is called a theorem 2 circle geometry.
(Image will be Uploaded Soon)
In the above figure, OD ⊥ AB, as per the theorem, hence, AD = DB.
(Image will be Uploaded Soon)
Proof:
Consider two triangles, ∆BOD and ∆AOD.
∠ADO = 90°, ∠BDO = 90° (AB ⊥OD) ………(Equation 1)
OB = OA (Radius of circle) ……….(Equation 2)
OD = OD (Common side) ………….(Equation 3)
From equation (1), equation (2) and equation (3);
∆AOB ≅ ∆POQ (Axiom of congruence R.H.S)
Hence, AD = DB (through CPCT rule)
Theorem 2 Circle Geometry: Converse Rule
A line segment that passes through the circle’s center bisects the chord and will be perpendicular to the chord.
(Image will be Uploaded Soon)
Proof: Circle Geometry
Consider ∆BOD and ∆AOD,
DB = AD (OD is a bisector of AB) ……….(Equation 1)
OA = OB (Radius) ……….( Equation 2)
OD = OD (Common side) ………..( Equation 3)
From equation 1, equation 2 and equation 3;
∆AOB ≅ ∆POQ (Axiom of congruence By SSS)
Hence,
∠ADO = ∠BDO = 90° (By CPCT rule)
Theorem 3:
Equal chords of the given circle are equidistant i.e. at equal distance from the circle’s center
Construction: Join OB and OD
(Image will be Uploaded Soon)
Proof:
Consider ∆OQD and ∆OPB.
BP = 1/2 AB (Perpendicular bisects the chord)…..(equation 1)
AB = CD (given in the theorem statement)
DQ = 1/2 CD (Perpendicular bisects the chord) …..(equation 2)
BP = DQ (from equation 1 and equation 2)
OB = OD (Radius)
∠OQD= ∠OPB = 90° (OQ ⊥ CD and OP ⊥ AB)
∆OPB ≅ ∆OQD (Axiom of Congruency, R.H.S)
Hence,
OP = OQ ( By CPCT rule)
Theorem 3: Converse Rule
Two chords of the circle that are at an equal distance from the circle’s center have the same length.
(Image will be Uploaded Soon)
Proof:
Consider ∆OQD and ∆OPB.
OQ = OP ………….(equation 1)
∠OPB = ∠OQD = 90° ………..(equation 2)
OB = OD (Radius) ……..(equation 3)
Hence, from equation 1, equation two and equation 3;
∆OPB ≅ ∆OQD (Axiom of Congruence By R.H.S)
BP = DQ (By CPCT rule)
1/2 AB = 1/2 CD (Perpendicular bisects the chord)
Hence,
AB = CD
Theorem 4:
The four points lie on a circle if a line segment connecting two points subtends equal angles at two other points on the same side of the line containing the line segment. (i.e. they are concyclic).
(Image will be Uploaded Soon)
Given: Let ABCD be a cyclic quadrilateral
We have to prove ∠A + ∠C = 180° and ∠B + ∠D = 180°
Construction: Join OB and OD.
Proof:
∠BOD = 2∠BAD
∠BAD = 1/2 ∠BOD
Similarly, ∠BCD = 1/2 reflex ∠BOD
∴ ∠BAD + ∠BCD = 1/2 ∠BOD + 1/2 reflex ∠BOD
= 1/2 (∠BOD + reflex ∠BOD) = 1/2 ×360°
∴ ∠A + ∠C = 180°
Similarly, ∠B + ∠D = 180°
Solved Example
Example 1
In the given figure of below circle, c is the center of the
circle.
(Image will be Uploaded Soon)
The circle’s diameter is BD.
A is the given point on the circle.
Find the angle CBA?
Answer: Diameter is given, BD
Angle BAD is confined within the circle (given).
Angle BAD =90°.
To find the angle CBA,
CBA =180−(23−90) = 67°
CBA = 67°
Example 2
Two circles with radii 5 cm and 3 cm intersect at two points. Their centers are 4 cm in distance. Calculate the common chord’s length.
(Image will be Uploaded Soon)
Answer: Given:
OP = 5cm
OS = 4cm and
PS = 3cm
Also, PQ = 2PR
Now, suppose RS = x.
Consider the ΔPOR,
OP2 = OR2+PR2
⇒ 52= (4-x)2+PR2
⇒ 25 = 16+x2-8x+PR2
∴ PR2 = 9-x2+8x — (i)
Now consider ΔPRS,
PS2 = PR2+RS2
⇒ 32 = PR2+x2
∴ PR2 = 9-x2 — (ii)
After equating both the equations, i.e. (i) and (ii). You will get,
9 -x2+8x = 9-x2
⇒ 8x = 0
⇒ x = 0
Now, in equation (i), put the value of x
PR2 = 9-02
⇒ PR = 3cm
∴ The length of the cord i.e. PQ = 2PR
So, PQ = 2×3 = 6cm
Conclusion
We hope this article on circle theorems helped the students to gain knowledge. Circle theorems are one of the important chapters. Practicing the theorems will help you grab hold of the concepts of all the theorems in a better way.
Thank you so much for taking the time to read the article titled What are Circle Theorems? written by Math Hello Kitty. Your support means a lot to us! We are glad that you found this article useful. If you have any feedback or thoughts, we would love to hear from you. Don’t forget to leave a comment and review on our website to help introduce it to others. Once again, we sincerely appreciate your support and thank you for being a valued reader!
Source: Math Hello Kitty
Categories: Math
