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Inscribed Angle of a Circle

A circle is a very important two-dimensional figure that is probably most commonly seen in geometry. There are a number of concepts and theorem related to a circle. A circle may form many different types of angles within it. The inscribed angle is one of them. This angle is formed by two chords or two secants within a circle. An inscribed angle is formed when two secant lines intersect on a circle. In other words, the inscribed angle in a circle is formed when one of the end points of two chords in a circle meet in a point.

inscribed angle

In the above picture, $\angle$ ABC is the inscribed angle of the circle. Let us go ahead and learn more about inscribed circle, such as - its definition, theorem based on it, example problems on this concept.

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What is Inscribed Angle ?

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The inscribed angle is demonstrated in the following diagram of a circle.

Example on Inscribed Angle

From the picture above,

  • Angle PQR is the inscribed angle.
  • PQ and QR are the chords.
  • Arc PR is the intercepted arc.
  • Formula: Angle PQR = $\frac{Arc PR}{2}$.

Inscribed Angle of a Circle

  • Two or more inscribed angles intercepting a same arc will be equal.

$\angle$ BAC = $\angle$BDC (From above figure)

  • An inscribed angle is the measure of half the central angle intercepting the same arc.

$\angle$ CAB = $\frac{1}{2}$ $\angle$BOC

$\angle$ CDB = $\frac{1}{2}$  $\angle$ BOC

Another Formula for inscribed angle:

If we know the length of the minor arc, radius, the inscribed angle is found by:

Angle=  $\frac{90L}{\pi \times R}$

where,

L is the length of the shortest arc BA

R is defined as radius.

Relationship to Thales' Theorem

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In geometry, according to the Thales' theorem if there are three points of a circle where line joining any two points is a diameter of circle, then angle formed by this diameter with third point on circle is a right angle. This theorem is said to be the special case of inscribed angle theorem.
Have a look at the following diagram :

Thales' Theorem

Given that points A and B are endpoints of a diameter of circle. Thales' theorem states that the inscribed angle would be a right angle in this case. We can prove this in the following way.

Let us suppose that $\angle$CAO = x and $\angle$CBO = y.
So, we would have $\angle$C = x + y
Sum of all three angles of a triangle is 180$^{\circ}$, therefore
x + y + (x + y) = 180$^{\circ}$ 
2(x + y) = 180$^{\circ}$
x + y = 90$^{\circ}$
Hence, it is proved.

Examples

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Example 1: From the circle diagram below, the chord CA has a length of 12 cm and center at O. The circle has a radius of 14 cm. Find the measure of the inscribed angle ABC.

Inscribed Angle Examples

Solution:

1. First we calculate the central angle AOC.The triangle AOC is aisosceles triangle. Distance of OC = Distance of OA = radius = 14 cm. Here we use cosine law to find value of cos (angle AOC).

$AC^2$ = $OC^2 + OA^2 - 2 OC\ OA \cos(\angle COA)$

2. Substitute the value ofthe angles AC, OC and AO in cos (angle AOC) as follows

$cos(\angle COA)$ $\frac{14^2 + 14^2 - 122}{28 \times 14}$

$\frac{62}{98}$

3. The measure of the angle COA is given by.

$\angle COA$ $\cos^{-1}$ $\frac{62}{98}$

According to the theorem described before, the size of angle CBA will be equal to half the size of angle COA.

$\angle CBA$ = $\frac{1}{2}$ $\cos^{-1}$ $\frac{62}{98}$ 

= 25.38 degrees.

Practice Problems

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Problem 1:

Given that the central angle of circle is 50 degrees. Find the angle of inscribed circle.

Answer = 100 degree

Problem 2:

Given that the central angle of circle is 120 degrees. Find the angle of inscribed circle.

Answer = 240 degree.

More topics in Inscribed Angle
Inscribed Angle Theorem
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