### All ISEE Upper Level Math Resources

## Example Questions

### Example Question #1 : Sectors

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . If the screen has a radius of 4 inches, what is the area of the highlighted wedge?

**Possible Answers:**

**Correct answer:**

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . If the screen has a radius of 4 inches, what is the area of the highlighted wedge?

To begin, let's recall our formula for area of a sector.

Now, we have theta and r, so we just need to plug them in and simplify!

So our answer is

### Example Question #1 : Sectors

A giant clock has a minute hand six feet long. How far, in inches, did the tip move between noon and 1:20 PM?

**Possible Answers:**

**Correct answer:**

The distance that the tip of the minute hand moves during one hour is the circumference of a circle with radius 6 feet. This circumference is feet. One hour and twenty minutes is hours, so the tip of the hand moved feet, or inches.

### Example Question #1 : Sectors

A giant clock has a minute hand three feet long. How far, in inches, did the tip move between noon and 12:20 PM?

**Possible Answers:**

It is impossible to tell from the information given

**Correct answer:**

The distance that the tip of the minute hand moves during one hour is the circumference of a circle with radius feet. This circumference is feet. minutes is one-third of an hour, so the tip of the minute hand moves feet, or inches.

### Example Question #1 : How To Find The Length Of An Arc

In the above figure, express in terms of .

**Possible Answers:**

**Correct answer:**

The measure of an arc - - intercepted by an inscribed angle - - is twice the measure of that angle, so

### Example Question #4 : How To Find The Length Of An Arc

In the above diagram, radius .

Give the length of .

**Possible Answers:**

**Correct answer:**

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

.

, being an inscribed angle of the circle, intercepts an arc with twice its measure:

The length of is the circumference multiplied by :

.

### Example Question #1 : Sectors

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . If the screen has a radius of 4 inches, what is the length of the arc of the highlighted wedge?

**Possible Answers:**

**Correct answer:**

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . If the screen has a radius of 4 inches, what is the length of the arc of the highlighted wedge?

To begin, let's recall our formula for length of an arc.

Now, just plug in and simplify

So, our answer is 4.54in

### 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?

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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 #2 : How To Find The Angle Of A Sector

In the above diagram, radius .

Calculate the length of .

**Possible Answers:**

**Correct answer:**

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 .

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