Arcs and Subtended Angles

Arcs and Subtended Angles

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How do you Find the Angle Subtended by an Arc in Circle?

An arc of a circle is any part of the circumference.

The angle subtended by an arc at any point is the angle formed between the two line segments joining that point to the end-points of the arc.

In the following figure, an arc of the circle shown subtends an angle \(\alpha\) at a point on the circumference, and an angle \(\beta\) at the centre \(O\)

Subtended angles: an arc of the circle shown subtends an angle alpha at a point on the circumference, and an angle beta at the center O

We are interested in the relation between the angle which a given arc subtends anywhere on the circumference and the angle which that arc subtends at the circle’s centre.

The relation between them is simple, and extremely important.


Angle Subtended by an Arc at the Centre

Theorem:

The angle subtended by an arc of a circle at its centre is twice of the angle it subtends anywhere on the circle’s circumference.

The proof of this theorem is quite simple, and uses the exterior angle theorem – an exterior angle of a triangle is equal to the sum of the opposite interior angles.

If the two opposite interior angles happen to be equal, then the exterior angle will be twice of any of the opposite interior angles.

Proof:

Consider the following figure, in which an arc (or segment) \(AB\) subtends \(\angle AOB\) at the centre \(O\), and \(\angle ACB\) at a point \(C\) on the circumference.

Angle Subtended by an Arc at the Centre of a circle

We have to prove that \(\angle AOB = 2 \times \angle ACB\). Draw the line through \(O\) and \(C\), and let it intersect the circle again at \(D\), as shown.

Angle Subtended by an Arc at the Centre of a circle

We make the following observations.

1. In \(\Delta OAC\), \(\angle OAC = \angle OCA\), because \(OA = OC\)

2. In \(\Delta OBC\), \(\angle OBC = \angle OCB\), because \(OB = OC\)

Hence,

\(\angle AOD = 2 \times \angle ACO \;\;\;\; \cdots (1)\)

\(\angle DOB = 2 \times \angle OCB \;\;\;\; \cdots (2)\)

Add equations \((1)\) and \((2)\):

\(\angle AOD + \angle DOB = 2 \times (\angle ACO + \angle OCB)\)

\(\implies \angle AOB = 2 \times \angle ACB\)

This completes the proof of the theorem.

You can experiement with the simulation below to understand the relationship between the angles subtended by an arc at the centre and at a point on the circle.

What if arc \(AB\) is such that it subtends a reflex angle at the centre?

Well, our proof doesn’t change, as the following figure shows.

Arc in a circle subtends reflex angle

You can now apply exactly the same steps as we did earlier, and arrive at the same result: \(\angle AOB = 2 \times \angle ACB\) 

Another configuration would be when \(O\) does not lie within \(\angle ACB\)

Will the result still hold, as in the following figure?

Arc not subtending an angle in a circle at the center

Yes, it will.

Join \(C\) to \(O\) again and extend it to \(D\) on the circumference of the circle.

Arc not subtending an angle at the center of a circle

Use the Exterior Angle Theorem again and note that,

1. \(\angle BOD = 2 \times \angle BCD \;\;\;\; \cdots (1)\)

2. \(\angle AOD = 2 \times \angle ACD \;\;\;\; \cdots (2)\)

To avoid cluttering, these angles have not been highlighted in the figure, but make sure you observe them carefully.

Subtract Equation \((2)\) from \((1)\)

\(\angle BOD - \angle AOD = 2 \times (\angle BCD - \angle ACD)\)

\(\implies \angle AOB = 2 \times \angle ACB\)

This theorem leads to an interesting corollary, which is discussed next.

Corollary:

The angle in a semi-circle is a right angle.

Consider the figure below, where \(AB\) is a diameter of the circle.

We need to prove that \(\angle ACB = 90^{\circ}\)

Angle in semi-circle is right angle.

Proof:

Since \(\angle AOB =180^{\circ}\),

\(\angle ACB = \begin{align}\frac{1}{2}\end{align} \times \angle AOB = 90^{\circ}\)

This theorem also leads to the following result.


Angles in the Same Segment of a Circle are Equal

Theorem:

Angles in the same segment of a circle are equal.

In other words, an arc in a circle will subtend equal angles anywhere on the circumference.

Consider the following figure, which shows an arc \(AB\) subtending angles \(ACB\) and \(ADB\) at two arbitrary points \(C\) and \(D\) on the circumference.

\(O\) is the centre of the circle.

Angles in same circle segment are equal.

We need to prove that \(\angle ACB = \angle ADB\)

Proof:

Using our previous theorem, we have that

\(\angle ACB = \begin{align}\frac{1}{2}\end{align} \times \angle AOB \;\;\;\; \cdots (1)\)

\(\angle ADB= \begin{align}\frac{1}{2}\end{align} \times \angle AOB \;\;\;\; \cdots (2)\)

From equations \((1)\) and \((2)\), we get

\(\angle ACB = \angle ADB\)

Now you can experiment with the simulation below to examine the relationship.

 
Thinking out of the box
Think Tank
  1. If the angles subtended by the chords of a circle at a centre are equal, are the chords equal?
  2. The chords of congruent circles subtend equal  angles at their centres. Are the chords equal?

Solved Examples

Example 1

 

 

Find the value of \(x\) (in degrees) in each of the following figures (wherever indicated, \(O\) is the centre of the circle)

I. 

Calculating third angle in circle using circle theorems

Solution:

From the figure, we can say that \(\angle ACB = \angle ADB = 40^{\circ}\)

Using the angle sum property in \(\Delta ACB\),

\(\begin{align}\angle BAC + \angle ACB + x^{\circ} &=180^{\circ}\\ 30^{\circ} + 40^{\circ} + x^{\circ}&=180^{\circ}\\x^{\circ}=110^{\circ}\end{align}\)

\(\therefore\) The value of \(x\) is 110

II. 

Calculating second angle in circle using circle theorems

Solution:

\(\angle BAD\) is an angle in a semi-circle, and is therefore, a right angle.

Using the Angle Sum Property in \(\Delta ABD\), we have:

\(\begin{align}\angle ADB&=180^{\circ}-(30^{\circ}+90^{\circ})\\&=60^{\circ}\end{align}\)

Thus, \(\angle ACB=\angle ADB=60^{\circ}\)

\(\therefore\) The value of \(x\) is 60

III.

Calculating exterior angle in circle using circle theorems

Solution:

We have \(\angle AOB=2\angle ACB\)

So,

\(\begin{align}\angle AOB &= 2\angle ACB\\ &=2(35^{\circ}+30^{\circ})\\ &=130^{\circ}\end{align}\)

\(\therefore\) The value of \(x\) is 130

IV.

Calculating interior angle in circle using circle theorems

Solution:

We note that,

\(\begin{align} \angle AOC&=360^{\circ}-(150^{\circ}+120^{\circ})\\&=90^{\circ}\end{align}\)

Thus, \(\angle ABC= \begin{align}\frac{1}{2}\end{align} \angle AOC=45^{\circ}\)

\(\therefore\) The value of \(x\) is 45

V.

Calculating of triangle's angle in circle using circle theorems

Solution:

We have, \(\angle BOC=180^{\circ}-40^{\circ}=140^{\circ}\)

\(\implies x^{\circ}= \begin{align}\frac{1}{2}\end{align} \angle BOC=70^{\circ}\)

\(\therefore\) The value of \(x\) is 70

VI.

Calculating parallelogram's angle in circle using theorem

Solution:

The reflex angle \(AOB\) is equal to \(360^{\circ}-150^{\circ}\) or \(210^{\circ}\)

Thus, \(x^{\circ}= \begin{align}\frac{1}{2}\end{align} \times 210^{\circ}=105^{\circ}\)

\(\therefore\) The value of \(x\) is 105
Example 2

 

 

In the following figure, find \(x^{\circ}\) in terms of \(y^{\circ}\) and \(z^{\circ}\)

calculating angles of triangles in circle using theorem

\(O\) is the centre of the circle.

Solution:

We label some angles as shown in the figure below.

calculating angles of triangles in circle using theorem

Using the exterior angle theorem, we have,

\(\angle y = \angle 1 + \angle 2 \;\;\;\; \cdots (1)\)

\(\angle 1 + \angle x = \angle 4 \;\;\;\; \cdots (2)\)

Add Equations \((1)\) and \((2\)

\(\angle y + \angle 1 + \angle x = \angle 1 + \angle 2 + \angle 4\)

\(\implies \angle x + \angle y = \angle 2 + \angle 4\)

Now, since \(z^{\circ} = 2 \angle 2 =2 \angle 4\), we have \(\angle 2 = \angle 4 = \begin{align}\frac{z^{\circ}}{2}\end{align}\) 

So,

\(\begin{align}x^{\circ}+y^{\circ}&=z^{\circ}\\ x^{\circ}&=z^{\circ}-y^{\circ}\end{align}\)

\(\therefore\) \(x^{\circ}=z^{\circ}-y^{\circ}\)
Example 3

 

 

Two circles intersect at \(P\) and \(Q\).

Through \(P\), diameters are drawn for both the circles.

The other end-points of these diameters are \(R\) and \(S\).

Show that \(R\), \(Q\) and \(S\) are collinear.

Solution:

Consider the following figure.

Two intersecting circles and two triangles

Since angles in semi-circles are \(90^{\circ}\), we have, \(\angle RQP=\angle SQP=90^{\circ}\)

Then,

\(\begin{align}\angle RQS&=\angle RQP+\angle SQP\\&=90^{\circ}+90^{\circ}\\&=180^{\circ}\end{align}\)

Thus, \(RQS\) is a straight line, which means that \(R\), \(Q\) and \(S\) collinear.

\(\therefore\) The points \(R\), \(Q\) and \(S\) are collinear.
Example 4

 

 

\(\Delta ABC\) is isosceles, with vertex \(A\)

Show that the circle through diameter \(AB\) bisects \(BC\) at point \(D\)

Solution:

Observe the following figure.

The circle cuts \(BC\) at \(D\)

circle and triangle

Since angles in a semi-circle are of measure \(90^{\circ}\), \(\angle ADB=90^{\circ}\)

This means \(AD\) is the altitude for the isosceles \(\Delta ABC\)

\(\implies\) \(AD\) also bisects \(BC\)

\(\implies\) \(D\) is the mid-point of \(BC\)
Example 5

 

 

From the end-points of a diameter \(AB\) of a circle with centre \(O\), perpendiculars \(AP\) and \(BQ\) are dropped on an external line.

A circle with a triangle inside it. AB is the diameter. O is the centre. Perpendiculars AP and BQ are dropped on an external line.

Show that \(AP = CQ\)

Solution:

We note that \(\angle ACB = 90^{\circ}\)

Thus, \(ACQP\) is a rectangle.

Since oppoite sides of a rectangle are equal, \(AP = CQ\)

Hence, proved that \(AP = CQ\)
 
important notes to remember
Important Notes
  1. Congruent arcs of a circle subtend equal angles at the centre.
  2. The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
  3. Angles in the same segment of a circle are equal.
  4. An angle in a semicircle is a right angle.

Practice Questions

Here are few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

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Frequently Asked Questions (FAQs)

1. What is the meaning of subtended?

In Geometry, subtended means to extend under.

When a line subtends an angle, then it forms an angle at the point where they meet.

2. What is a subtended central angle?

A subtended central angle is the angle formed at the centre of the circle and the legs of the angle are the radii of the circle intersecting the circle at two different points.

3. What is the angle subtended by a chord at a point?

The angle subtended by a chord is opposite to the chord and the arms of the angle meet the end points of the chord.

In the following figure, chord \(AB\) subtends an angle \(\angle AOB\) and chord \(CD\) subtends an angle \(\angle COD\)

In a circle, chord AB subtends an angle AOB and chord CD subtends an angle COD

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