SAT Math : How to find the length of a chord

Study concepts, example questions & explanations for SAT Math

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Example Questions

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

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 : How To Find The Length Of A Chord

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 #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:

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 #4 : 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 #2 : 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:

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 #3 : Circles

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.

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