# ISEE Upper Level Math : Circles

## Example Questions

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### Example Question #5 : How To Find The Angle Of A Sector

Figure NOT drawn to scale.

The circumference of the above circle is 120.  and  have lengths 10 and 20, respectively. Evaluate .

Possible Answers:

Correct answer:

Explanation:

The length of   comprises  of the circumference of the circle. Therefore, its degree measure is  . Similarly, The length of   comprises  of the circumference of the circle. Therefore, its degree measure is

If two secants are constructed to a circle from an outside point, the degree measure of the angle the secants form is half the difference of those of the arcs intercepted - that is,

.

### Example Question #6 : How To Find The Angle Of A Sector

Figure NOT drawn to scale.

Refer to the above diagram.  and  have lengths 80 and 160, respectively. Evaluate .

Possible Answers:

Correct answer:

Explanation:

The circumference of the circle is the sum of the two arc lengths:

The length of   comprises  of the circumference of the circle. Therefore, its degree measure is  . Consequently,  is an arc of degree measure

The segments shown are both tangents from  to the circle. Consequently, the degree measure of the angle they form is half the difference of the angle measures of the arcs they intercept - that is,

### Example Question #7 : How To Find The Angle Of A Sector

Figure NOT drawn to scale.

The circumference of the above circle is 100.  and  have lengths 20 and 15, respectively. Evaluate .

Possible Answers:

Correct answer:

Explanation:

The length of   comprises  of the circumference of the circle. Therefore, its degree measure is  . Similarly, The length of   comprises  of the circumference of the circle. Therefore, its degree measure is

If two chords cut each other inside the circle, as  and  do, and one pair of vertical angles are examined, then the degree measure of each angle is half the sum of those of the arcs intercepted - that is,

### Example Question #1 : How To Find The Percentage Of A Sector From An Angle

Sector SOW has a central angle of . What percentage of the circle does it cover?

Possible Answers:

Correct answer:

Explanation:

Sector SOW has a central angle of . What percentage of the circle does it cover?

Recall that there is a total of 360 degrees in a circle. SOW occupies 45 of them. To find the percentage, simply do the following:

### Example Question #1 : How To Find The Percentage Of A Sector From An Angle

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . What percentage of the circle is highlighted?

Possible Answers:

Correct answer:

Explanation:

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . What percentage of the circle is highlighted?

To find the percentage of a sector, simply put the degree measure of the angle over 360 and multiply by 100.

So, our answer is 18.06%

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