# ISEE Upper Level Quantitative : Circles

## Example Questions

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### Example Question #1 : Circles

The clock in the classroom reads 5:00pm. What is the angle that the hands are forming?

Explanation:

Since the clock is a circle, you can determine that the total number of degrees inside the circle is 360. Since a clock has 12 numbers, we can divide 360 by 12 to see what the angle is between two numbers that are right next to each other. Thus, we can see that the angle between two numbers right next to each other is . However, the clock is reading 5:00, so there are five numbers we have to take in to account. Therefore, we multiply 30 by 5, which gives us as our answer.

### Example Question #2 : Circles

The time on a clock reads 5:00. What is the measure of the central angle formed by the hands of the clock?

Explanation:

First, remember that the number of degrees in a circle is 360. Then, figure out how many degrees are in between each number on the face of the clock. Since there are 12 numbers, there are between each number. Since the time reads 5:00, multiply , which yields .

### Example Question #3 : Circles

Refer to the above figure. Which is the greater quantity?

(a)

(b) 3

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

Explanation:

If two chords intersect inside a circle, both chords are cut in a way such that the products of the lengths of the two chords formed in each are the same - in other words,

or

Therefore, .

### Example Question #4 : Circles

Figure NOT drawn to scale

In the above figure,  is the center of the circle, and  is a tangent to the circle. Also, the circumference of the circle is .

Which is the greater quantity?

(a)

(b) 25

(a) and (b) are equal

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

Explanation:

is a radius of the circle from the center to the point of tangency of , so

,

and  is a right triangle. The length of leg  is known to be 24. The other leg  is a radius radius; we can find its length by dividing the circumference by :

The length hypotenuse, , can be found by applying the Pythagorean Theorem:

.

### Example Question #5 : Circles

Figure NOT drawn to scale.

Refer to the above figure. Which is the greater quantity?

(a)

(b) 7

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

(b) is the greater quantity

Explanation:

If two chords intersect inside a circle, both chords are cut in a way such that the products of the lengths of the two chords formed in each are the same - in other words,

Solving for :

Since , it follows that , or .

### Example Question #6 : Circles

In the above figure,  is a tangent to the circle.

Which is the greater quantity?

(a)

(b) 32

(a) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(b) is the greater quantity

(b) is the greater quantity

Explanation:

If a secant segment and a tangent segment are constructed to a circle from a point outside it, the square of the distance to the circle along the tangent is equal to the product of the distances to the two points on the circle along the secant; in other words,

Simplifying, then solving for :

To compare  to 32, it suffices to compare their squares:

, so, applying the Power of a Product Principle, then substituting,

, so

it follows that

.

### Example Question #7 : Circles

Figure NOT drawn to scale

In the above figure,  is a tangent to the circle.

Which is the greater quantity?

(a)

(b) 8

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

(a) and (b) are equal

Explanation:

If a secant segment and a tangent segment are constructed to a circle from a point outside it, the square of the distance to the circle along the tangent is equal to the product of the distances to the two points on the circle intersected by the secant; in other words,

Simplifying and solving for :

Factoring out :

Either  - which is impossible, since  must be positive, or

, in which case .

### Example Question #8 : Circles

Refer to the above figure. Which is the greater quantity?

(a)

(b)

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

Explanation:

If two chords intersect inside a circle, both chords are cut in a way such that the products of the lengths of the two chords formed in each are the same - in other words,

Divide both sides of this equation by , then cancelling:

The two quantities are equal.

### Example Question #9 : Circles

The area of Circle B is four times that of Circle A. The area of Circle C is four times that of Circle B. Which is the greater quantity?

(a) Twice the radius of Circle B

(b) The sum of the radius of Circle A and the radius of Circle C

(a) is greater.

(b) is greater.

It cannot be determined from the information given.

(a) and (b) are equal.

(b) is greater.

Explanation:

Let  be the radius of Circle A. Then its area is .

The area of Circle B is , so the radius of Circle B is twice that of Circle A; by a similar argument, the radius of Circle C is twice that of Circle B, or .

(a) Twice the radius of circle B is .

(b) The sum of the radii of Circles A and B is .

This makes (b) greater.

### Example Question #10 : Circles

The time is now 1:45 PM. Since noon, the tip of the minute hand of a large clock has moved  feet. How long is the minute hand of the clock?

Explanation:

Every hour, the tip of the minute hand travels the circumference of a circle. Between noon and 1:45 PM, one and three-fourths hours pass, so the tip travels  or  times this circumference. The length of the minute hand is the radius of this circle , and the circumference of the circle is , so the distance the tip travels is  this, or

Set this equal to  feet:

feet.

This is equivalent to 1 foot 4 inches.

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