ISEE Upper Level Math : How to find the angle of a sector

Example Questions

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

A giant clock has a minute hand four feet long. Since noon, the tip of the minute hand has traveled  feet. What time is it now?

Explanation:

The circumference of the path traveled by the tip of the minute hand over the course of one hour is:

feet.

Since the tip of the minute hand has traveled  feet since noon, the minute hand has made

revolutions. Therefore,  hours have elapsed since noon, making the time 1:15 PM.

Example Question #2 : How To Find The Angle Of A Sector

Figure NOT drawn to scale

Refer to the above diagram. is a semicircle. Evaluate  given .

Explanation:

An inscribed angle of a circle that intercepts a semicircle is a right angle; therefore, , which intercepts the semicircle , is such an angle. Consequently,  is a right triangle, and  and  are complementary angles. Therefore,

Inscribed  intercepts an arc with twice its angle measure; this arc is , so

.

The major arc corresponding to this minor arc, , has measure

Example Question #3 : How To Find The Angle Of A Sector

Note: Figure NOT drawn to scale

Refer to the above diagram. is a semicircle. Evaluate .

Explanation:

An inscribed angle of a circle that intercepts a semicircle is a right angle; therefore, , which intercepts the semicircle , is such an angle. Consequently,

Inscribed  intercepts an arc with twice its angle measure; this arc is , so

.

Example Question #4 : How To Find The Angle Of A Sector

In the above diagram, radius .

Calculate the length of .

Explanation:

Inscribed , which measures , intercepts an arc with twice its measure. That arc is , which consequently has measure

.

This makes  an arc which comprises

of the circle.

The circumference of a circle is  multiplied by its radius, so

.

The length of  is  of this, or .

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 .

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 .

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 .

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,