SAT Math : Circles

Study concepts, example questions & explanations for SAT Math

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Example Questions

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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 .

Explanation:

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

Chords

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 : Chords

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:

Explanation:

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

Chords

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 #3 : Chords

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:

Explanation:

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

The figure referenced is below:

Chords

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 #1 : How To Find The Length Of A Chord

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:

Explanation:

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

Chords

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 :

which is the length of .

Example Question #5 : Chords

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:

Explanation:

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.

Chords

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,

Set  and , and ; substitute and solve for :

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

Example Question #6 : Chords

Secant

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

Explanation:

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

If is 2:00 PM, what is the measure of the angle between the minute and hour hands of the clock?

Possible Answers:

60 degrees

30 degrees

45 degrees

120 degrees

90 degrees

Correct answer:

60 degrees

Explanation:

First note that a clock is a circle made of 360 degrees, and that each number represents an angle and the separation between them is 360/12 = 30. And at 2:00, the minute hand is on the 12 and the hour hand is on the 2.  The correct answer is 2 * 30 = 60 degrees.

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:

Explanation:

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 #251 : Plane Geometry

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

Possible Answers:

Correct answer:

Explanation:

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 #2 : 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:

30°

120°

45°

90°

56°

Correct answer:

120°

Explanation:

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).

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