### All SAT Math Resources

## Example Questions

### Example Question #1 : Circles

Two chords of a circle, and , intersect at a point . is twice as long as , , and .

Give the length of .

**Possible Answers:**

Insufficient information is given to find the length of .

**Correct answer:**

Insufficient information is given to find the length of .

Let stand for the length of ; then the length of is twice this, or . The figure referenced is below:

If two chords intersect inside the circle, then they cut each other in such a way that the product of the lengths of the parts is the same for the two chords - that is,

Substituting the appropriate quantities, then solving for :

This statement is identically true. Therefore, without further information, we cannot determine the value of - the length of .

### Example Question #2 : Circles

Two chords of a circle, and , intersect at a point . is 12 units longer than , , and .

Give the length of (nearest tenth, if applicable)

**Possible Answers:**

**Correct answer:**

Let stand for the length of ; then the length of is . The figure referenced is below:

If two chords intersect inside the circle, then they cut each other in such a way that the product of the lengths of the parts is the same for the two chords - that is,

Substituting the appropriate quantities, then solving for :

This quadratic equation can be solved by completing the square; since the coefficient of is 12, the square can be completed by adding

to both sides:

Restate the trinomial as the square of a binomial:

Take the square root of both sides:

or

Either

,

in which case

,

or

in which case

,

Since is a length, we throw out the negative value; it follows that , the correct length of .

### Example Question #1 : How To Find The Length Of A Chord

A diameter of a circle is perpendicular to a chord at a point .

What is the diameter of the circle?

**Possible Answers:**

Insufficient information is given to answer the question.

**Correct answer:**

In a circle, a diameter perpendicular to a chord bisects the chord. This makes the midpoint of ; consequently, .

The figure referenced is below:

If two chords intersect inside the circle, then they cut each other in such a way that the product of the lengths of the parts is the same for the two chords - that is,

Setting , and solving for :

,

the correct length.

### Example Question #4 : Circles

Two chords of a circle, and , intersect at a point .

Give the length of .

**Possible Answers:**

Insufficient information is given to answer the question.

**Correct answer:**

Let , in which case ; the figure referenced is below (not drawn to scale).

Setting , and solving for :

,

which is the length of .

### Example Question #1 : Circles

A diameter of a circle is perpendicular to a chord at point . and . Give the length of (nearest tenth, if applicable).

**Possible Answers:**

insufficient information is given to determine the length of .

**Correct answer:**

A diameter of a circle perpendicular to a chord bisects the chord. Therefore, the point of intersection is the midpoint of , and

.

Let stand for the common length of and ,

The figure referenced is below.

Set and , and ; substitute and solve for :

This is the length of ; the length of is twice this, so

### Example Question #6 : Circles

Figure is not drawn to scale

In the provided diagram, the ratio of the length of to that of is 7 to 2. Evaluate the measure of .

**Possible Answers:**

Cannot be determined

**Correct answer:**

Cannot be determined

The measure of the angle formed by the two secants to the circle from a point outside the circle is equal to half the difference of the two arcs they intercept; that is,

The ratio of the degree measure of to that of is that of their lengths, which is 7 to 2. Therefore,

Letting :

Therefore, in terms of :

Without further information, however, we cannot determine the value of or that of . Therefore, the given information is insufficient.

### Example Question #1 : How To Find The Angle Of Clock Hands

It is 4 o’clock. What is the measure of the angle formed between the hour hand and the minute hand?

**Possible Answers:**

**Correct answer:**

At four o’clock the minute hand is on the 12 and the hour hand is on the 4. The angle formed is 4/12 of the total number of degrees in a circle, 360.

4/12 * 360 = 120 degrees

### Example Question #1 : How To Find The Angle Of Clock Hands

If a clock reads 8:15 PM, what angle do the hands make?

**Possible Answers:**

**Correct answer:**

A clock is a circle, and a circle always contains 360 degrees. Since there are 60 minutes on a clock, each minute mark is 6 degrees.

The minute hand on the clock will point at 15 minutes, allowing us to calculate it's position on the circle.

Since there are 12 hours on the clock, each hour mark is 30 degrees.

We can calculate where the hour hand will be at 8:00.

However, the hour hand will actually be *between* the 8 and the 9, since we are looking at 8:15 rather than an absolute hour mark. 15 minutes is equal to one-fourth of an hour. Use the same equation to find the additional position of the hour hand.

We are looking for the angle between the two hands of the clock. The will be equal to the difference between the two angle measures.

### Example Question #1 : How To Find The Angle Of Clock Hands

What is the measure of the smaller angle formed by the hands of an analog watch if the hour hand is on the 10 and the minute hand is on the 2?

**Possible Answers:**

120°

30°

45°

90°

56°

**Correct answer:**

120°

A analog clock is divided up into 12 sectors, based on the numbers 1–12. One sector represents 30 degrees (360/12 = 30). If the hour hand is directly on the 10, and the minute hand is on the 2, that means there are 4 sectors of 30 degrees between then, thus they are 120 degrees apart (30 * 4 = 120).

### Example Question #1 : How To Find The Angle Of Clock Hands

At , what angle is between the hour and minute hand on a clock?

**Possible Answers:**

**Correct answer:**

At , the hour hand is on the and the minute hand is at the . There are spaces on a clock, and these hands are separated by spaces.

Thus, the angle between them is the degrees of the entire clcok, which is .

Therefore, we multiply these to get our answer.

We can cancel out as we multiply to get:

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