Circle Theorems: Definition, Formulas, Steps, Examples, FAQs

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What Are Circle Theorems in Geometry?

The circle theorems in geometry are statements that prove significant results about circles. These theorems provide important information or facts about various parts of a circle. 

We can use circle theorems and previous knowledge of properties of a circle to calculate missing angles. The circle angle theorems give us the rules for finding the measurements of angles in a circle.

A circle is a locus of points that are equidistant from a fixed point called center. Different theorems are based on the various parts of a circle, such as radius, central angles, tangents, sectors, the chords, etc. These theorems help us to solve many problems in geometry easily.

Before we discuss the theorems, let’s revise the parts of the circle using a diagram.

Parts of a circle

Here are a few important terms that are commonly used in the statements of circle theorems.

Subtended angle: The angle made by something (arc, chord, etc) from a given viewpoint. An angle within a circle created by two chords meeting at a point on the circumference. In a circle, there are various angles that can be formed joining the endpoints of arcs and those angles are termed subtended angles.

Inscribed Angle: An angle made from points lying on the circle’s circumference. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. Thus, it is formed by two chords that have a common endpoint on the circle. 

Central Angle: An angle whose vertex is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points.

Important terms in circle theorems

Circle Theorems

We will discuss the following circle theorems in detail.

  1. Alternate segment theorem
  2. The angle at the center theorem
  3. Angles in the same segment theorem
  4. Angle in a semicircle theorem
  5. Chord of a circle theorem
  6. Cyclic quadrilateral theorem
  7. Tangent of a circle theorem-1
  8. Tangent of a circle theorem-2

Circle Theorem 1: Alternate Segment Theorem

Alternate Segment Theorem: For any circle, the angle formed between the tangent and the chord through the point of contact of the tangent is equal to the angle formed by the same chord in the alternate segment.

Alternate segment theorem visual

Here, according to the alternate segment theorem, 

  • ∠x = ∠BCE = ∠BAC

Here, ∠BCE is the angle made by the tangent and the chord BC at the point of contact.

∠BAC is the angle made by the same chord BC in the alternate segment BAC.

  • Similarly, ∠y = ∠ABC = ∠ACD

Circle Theorem 2: Central Angle Theorem (The Angle at the Center Theorem)

Central Angle Theorem: The central angle subtended by two points on a circle is always twice the inscribed angle subtended by those points.

Here, the angle BAC is known as the inscribed angle, an angle made from points lying on the circle’s circumference.

The angle BOC is the central angle, an angle whose vertex is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points B and C.

Thus, the theorem states that an inscribed angle ∠BAC = x° is half of the measure of the central angle ∠BOC = 2x°.

∠BOC = 2 ∠BAC.

Angle at the center theorem

Circle Theorem 3: Angles in the Same Segment Theorem

Angles in the same segment theorem: Angles formed in the same segment (either major segment or minor segment) of a circle are equal.

Here, according to the angles in the same segment theorem: ∠BAD = ∠BCD

Angles in the same segment theorem

Circle Theorem 4: Angle in a Semicircle Theorem

Angle in a semicircle theorem: An angle in a semicircle is always a right angle (measures 90 degrees).

In other words, an angle subtended by a circle’s diameter is always a right angle.

Angle in a semicircle is always 90 degrees

Here, according to the angle in a semicircle theorem: ∠A = ∠B = ∠C = 900

Angles A, B, C in a semicircle are right angles

Circle Theorem 5: Chord of a Circle Theorem

Chord of a circle theorem: The perpendicular drawn from the center of the circle to a chord bisects the chord.

Here, according to the chord of a circle theorem, we have 

∠OPA = ∠OPB = 900

AP = PB

Chord of a circle theorem

Circle Theorem 6: Angles Subtended by Equal Chords Theorem

Angles subtended by equal chords theorem: Two equal chords of a circle subtend equal angles at the center of the circle.

The converse is also true. If two angles subtended at the center by two chords are equal, then the chords are of equal length.

In the given diagram, the chords XY and ST are equal. Thus, the angles subtended by them are also equal.

∠AOB = ∠POQ

Angles subtended by equal chords are equal.

Circle Theorem 7: Cyclic Quadrilateral Theorem

Cyclic quadrilateral theorem: The sum of opposite angles in a cyclic quadrilateral is equal to 1800.

A cyclic quadrilateral is a quadrilateral where all four vertices lie on a circle.

Cyclic quadrilateral ABCD

Here, ABCD is a cyclic quadrilateral. According to the cyclic quadrilateral theorem: 

∠A + ∠C = 1800 and ∠B + ∠D = 1800

Circle Theorem 8: Tangent of a Circle Theorem-1

Tangent of a circle theorem-1: The tangent makes a right angle with the radius of a circle at the point of tangency.

In the given diagram, the radius OP is perpendicular to the tangent AB. 

OP ⊥ AB.

∠OPA = ∠OPB = 900.

Tangent of a circle theorem

Circle Theorem 9: Two Tangents Theorem

If two tangent lines PA and PB are drawn from an external point, then

  • Two tangents have the same length. (PA = PB)
  • The two tangents subtend equal angles at the center of the circle. (∠POA = ∠POB)
  • The line that connects the exterior point to the center will divide the angle between the tangents into two equal angles. (∠OPA = ∠OPB)
Tangent of a circle theorem

Circle Theorems Formulas

TheoremsShapesFormula
Alternate Segment TheoremAlternate segment theorem∠x = ∠y 
and
∠a = ∠b
The central angle subtended by two points on a circle is always twice the inscribed angle subtended by those points.
Central angle theorem∠BOC = 2∠BAC
Angles in the same segment are congruent.angles in the same segment theorem∠PRQ = ∠PSQ
Angles in a semicircle are right angles.angles in a semicircle theorem∠A = ∠B = ∠C = 900
Chord of a circle theoremchord of a circle theorem AP = PB
∠OPA = ∠OPB = 900 
Angles subtended by equal chordsAngles subtended by equal chords∠XOY = ∠SOT
Cyclic quadrilateral theoremCyclic quadrilateral theorem∠A + ∠C = 1800 and ∠B + ∠D = 1800
Tangent of a circle theorem-1Tangent makes a right angle with the radius ∠OPA = ∠OPB = 900
Tangent of a circle theorem-2Tangent of a circle theorem PA = PB∠POA = ∠POB∠OPA = ∠OPB

Facts about Circle Theorems

  • The first theorems based on the circles are attributed to Thales around 650 BC.
  • In the fields of engineering and design, the circle theorems have important applications.

Conclusion

In this article, we delved into the essential circle theorems that provide valuable insights into angles, arcs, and their relationships within a circle. Understanding these theorems is crucial for various geometric applications. Now, let’s reinforce our knowledge by tackling examples and practicing MCQs for better comprehension.

Solved Examples on Circle Theorems

  1. In the circle given below, triangle ABC is inscribed in the circle and the tangent DE meets the circle at the point B. Find the measure of angle “x” and “y.”
Alternate segment theorem example

Solution: 

We know that the sum of interior angles of a triangle is equal to 180

∠BAC + ∠ACB + ∠ABC = 1800

x + 570480 = 1800

x = 18001050

x = 750

Now, according to the alternate segment theorem, the angle formed between the tangent and the chord through the point of contact of the tangent is equal to the angle formed by the same chord in the alternate segment.

x = y = 750

Hence, the measure of ∠x and ∠y is 750.

  1. In the figure given below, find the value of x using the circle theorems.
Angles in a semicircle theorem example

Solution:

In the given figure, ∠ABC is an angle in a semicircle. 

Angle in a semicircle is always a right angle. 

Thus, m∠ABC = 900

We know that the sum of interior angles of a triangle is equal to 1800

m∠BAC + m∠ACB + m∠ABC = 1800

x + 370 + 900 = 1800

x + 1270 = 1800

x = 18001270 = 530

Hence, the value of x is 530.

  1. In the figure given below, find the value of x using the circle theorems.
Example of angle at the center theorem

Solution: 

According to “the angle at the center theorem,” the central angle subtended by two points on a circle is always twice the inscribed angle subtended by those points.

∠AOB = 2∠ACB.

x = 2 × 470 = 940

The value of x is 940.

  1. In the given figure, the point O is a center of a circle, the radius of a circle is 17 inches and OP = 8 inches. Find the length of the chord AB.
Example of a chord of a circle theorem

Solution:

In the given figure, 

OA = OB = 17 inches 

OP = 8 inches

OP⊥ AB

Using pythagoras theorem in triangle OPB, we get

OP2 + PB2 = OB2

82 + PB2 = 172

PB2 = 172 − 82 = 225

Therefore, PB = $\sqrt{225}$ = 15 inches

According to the chord of a circle theorem, the perpendicular drawn from the center of the circle bisects the chord.

Thus, AP = PB

∴ AB = AP + PB = 15 inches + 15 inches = 30 inches.

Hence, the length of the chord AB is 30 inches.

  1. Find the value of “x” and “y” using the cyclic quadrilateral theorem.
Example on cyclic quadrilateral theorem

Solution:

In the given figure, ABCD is cyclic quadrilateral. 

We know that the sum of opposite angles in a cyclic quadrilateral is equal to 1800.

So, x + 980 = 1800

 x = 1800980 = 820

Similarly,

y + 950 = 1800

 x = 1800950 = 850

The values of x and y are 82 and 85 respectively.

  1. Find the value of “x” in the figure given below.
Tangent to a circle theorem example

Solution:

Two tangent lines PA and PB are drawn from an external point “P” as shown in the figure, then two tangents have the same length.

Thus, PA = PB.

x2 + 10 = 46

x2 = 46 − 10

x2 = 36

x = 6

Hence, the value of “x” is 6.

Practice Problems on Circle Theorems

Circle Theorems: Definition, Formulas, Steps, Examples, FAQs

Attend this quiz & Test your knowledge.

1

The angle subtended by a diameter at any point of the circumference is equal to _______.

$45^{\circ}$
$60^{\circ}$
$90^{\circ}$
$180^{\circ}$
CorrectIncorrect
Correct answer is: $90^{\circ}$
The angle subtended by a diameter at any point of the circumference is equal to $90^{\circ}$.
2

If the inscribed angle is $50^{\circ}$, then the central angle is ________.

$50^{\circ}$
$100^{\circ}$
$110^{\circ}$
$25^{\circ}$
CorrectIncorrect
Correct answer is: $100^{\circ}$
We know that the inscribed angle is half of the measure of the central angle.
Central angle $= 2 \times 50^{\circ} = 100^{\circ}$
3

The opposite angles in a cyclic quadrilateral are ______.

complementary
equal
supplementary
right angles
CorrectIncorrect
Correct answer is: supplementary
The opposite angles in a cyclic quadrilateral are supplementary. They add up to $180^{\circ}$.
4

Angles formed in the same segment of a circle are ______.

equal
different
right angles
none of these
CorrectIncorrect
Correct answer is: equal
Angles formed in the same segment of a circle are always equal.

Frequently Asked Questions on Circle Theorems

Geometry problems can be solved with the help of circle theorems. There are a number of useful patterns and theorems that can be deduced from drawing angles and lines inside a circle, which we can use in both practical and theoretical situations.

To find angles using the circle theorems based on a given condition and shape, we can apply circle theorems based on angles. Some important circle theorems based on angles are as follows:

  • The measure of angles subtended by an arc at the center of a circle is equal to twice the angles at any point on the circumference of the circle from the same arc.
  • Angles formed in the same segment of a circle are equal.
  • The angle in a semicircle is always 90 degrees.

The sum of opposite angles in a cyclic quadrilateral is equal to 1800.

The word circle is derived from the Greek word kirkos, meaning hoop or ring.

Following are the list of circle theorems:

  1. Alternate segment theorem
  2. The angle at the center theorem
  3. Angles in the same segment theorem
  4. Angle in a semicircle theorem
  5. Chord of a circle theorem
  6. Cyclic quadrilateral theorem
  7. Tangent of a circle theorems

A quadrilateral with all four vertices lying on a circle is known as a cyclic quadrilateral. Sometimes, It is also known as the inscribed quadrilateral. The sum of opposite angles in a cyclic quadrilateral is equal to 1800.

As we know that the sum of all angles of a quadrilateral is equal to 3600. Hence, the sum of all angles of a cyclic quadrilateral is also equal to 3600.