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# Intercepted Arc – Explanation & Examples

Now that we have learned all the basic parts of the circle let’s go into something complex. We are talking about the **intercepted arc**, **which is formed in the circle due to external lines**. If you are really good at angles, then this lesson should not be a problem for you to understand.

We saw all the basic definitions of parts of circles before, like diameter, chord, vertex, and central angle; if you have not, please go through the previous lessons because these parts have a use in this lesson.

**In this article, you will learn:**

- The definition of an intercepted arc,
- how to find an intercepted arc and,
- intercepted arc formula.

## What is an Intercepted Arc?

To recall, an arc is part of the circumference of a circle.** An intercepted arc can therefore be defined as an arc formed when one or two different chords or line segments cut across a circle and meet at a common point called a vertex. **

It is important to note that the lines or the chords can either meet in the middle of a circle, on the other side of a circle or outside a circle.

Or we can also define the intercepted arc as when two lines cross a circle at two different points, the part of the circle between the points of intersection forms the intercepted arc.

## How to Find Intercepted Arc?

There exist some interesting relationships between an intercepted arc and the inscribed and central angle of a circle. In geometry, an** inscribed angle** is formed between the chords or lines cutting across a circle.

**The central angle is an angle formed by two radii that joins the ends of a chord to the center of a circle**. These relationships between different intercepted arcs and their corresponding inscribed angles form the intercepted arc formula.

Let’s take a look.

### Intercepted arc formula

- Intercepted arc formula for lines meeting in the middle of a circle

**The central angle = the measure of the intercepted arc**

- Intercepted arc formula for chords meeting on the other side of a circle.

**The inscribed angle = 1/2 × intercepted arc**

Or

**2 x the inscribed angle = the intercepted arc**

Intersecting chords:

For intersecting chords, the intercepted arc is given by,

**The inscribed angle = half the sum of intercepted arcs.**

External inscribed angle:

**The size of the vertex angle outside the circle = 1/2 × (difference of intercepted arcs)**

Worked out examples about the intercepted arc.

*Example 1*

Find angle *ABC* in the circle shown below.

__Solution__

Given, the intercepted arc = 150°

The central angle = intercepted arc

Therefore, ∠*ABC* = 150°

*Example 2*

Determine the value of x in the circle shown below.

** **

__Solution__

The central angle = intercepted arc

60° = (3x + 15) °

Simplify

60° = 3x + 15°

Subtract 15° on both sides.

45° = 3x

Divide both sides by 3

x = 15°

So, the value of x is 15°.

*Example 3*

Find the value of the intercepted arc in the diagram shown below.

__Solution__

Given,

The inscribed angle = 15°

By the formula,

The inscribed angle = ½ × intercepted arc

15° = ½ x intercepted arc

Therefore, the measure of the intercepted arc is 30°.

*Example 4*

If the intercepted arc in the diagram below is 160°, determine the value of x.

__Solution__

Given,

The intercepted arc =160°

The inscribed angle = ½ × intercepted arc

The inscribed angle = ½ x 160°

= 80°

So, we have,

2(4x + 21) ° = 80°

8x + 42° = 80°

Subtract 42° on both sides.

8x = 38°

Divide both sides by 8 to get.

x = 4.75°

Thus, the value of x is 4.75°

*Example 5*

Find the value of the inscribed angle in the following diagram.

__Solution__

The inscribed angle = half the sum of intercepted arcs.

= ½ x (170° + 50°)

= ½ x 220°

= 110°

So, the inscribed angle is 110°.

*Example 6*

Find the value of x in the diagram shown below.

__Solution__

Given the intercepted arcs as 62° and 150°

The inscribed angle = half the sum of intercepted arcs.

The inscribed angle = ½ (62° + 150°)

= ½ x 212°

= 106°

Now solve for x.

(2x + 10) ° = 106°

Simplify.

2x + 10° =106°

Subtract 10° on both sides.

2x = 96

On dividing both sides by 2, we get,

x = 48°

Hence, the value of x is 48 degrees.

*Example 7*

Find the external vertex angle in the diagram shown below.

__Solution__

Now you need to recall the properties we studied above.

The size of the vertex angle outside the circle = 1/2 × (difference of intercepted arcs)

Vertex angle = ½ (140° – 40°)

= ½ x 100°

= 50°

So, the measure of angle with vertex outside the circle is 50°.

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