# ISEE Upper Level Math : How to find the length of a chord

## Example Questions

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

Which equation is the formula for chord length?

Note: is the radius of the circle, and   is the angle cut by the chord.

Explanation:

The length of a chord of a circle is calculated as follows:

Chord length =

### Example Question #2 : How To Find The Length Of A Chord

The radius of a circle is , and the perpendicular distance from a chord to the circle center is .  Give the chord length.

Explanation:

Chord length = , where   is the radius of the circle and   is the perpendicular distance from the chord to the circle center.

Chord length =

Chord length =

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

In the circle below, the radius is  and the chord length is . Give the perpendicular distance from the chord to the circle center (d).

Explanation:

Chord length = , where   is the radius of the circle and   is the perpendicular distance from the chord to the circle center.

Chord length =

### Example Question #4 : How To Find The Length Of A Chord

Give the length of the chord of a  central angle of a circle with radius 18.

Explanation:

The figure below shows , which matches this description, along with its chord :

By way of the Isoscelese Triangle Theorem,  can be proved a 45-45-90 triangle with legs of length 18, so its hypotenuse - the desired chord length  - is  times this, or .

### Example Question #2 : How To Find The Length Of A Chord

central angle of a circle intercepts an arc of length ; it also has a chord. What is the length of that chord?

Explanation:

The arc intercepted by a  central angle is  of the circle, so the circumference of the circle is . The radius is the circumference divided by , or

The figure below shows a  central angle , along with its chord :

By way of the Isoscelese Triangle Theorem,  can be proved equilateral, so .

### Example Question #6 : How To Find The Length Of A Chord

central angle of a circle intercepts an arc of length ; it also has a chord. What is the length of that chord?

Explanation:

The arc intercepted by a  central angle is  of the circle, so the circumference of the circle is . The radius is the circumference divided by , or

The figure below shows a  central angle , along with its chord and triangle bisector

We will concentrate on , which is a 30-60-90 triangle. By the 30-60-90 Theorem,

and

is the midpoint of , so

### Example Question #7 : How To Find The Length Of A Chord

Give the length of the chord of a  central angle of a circle with radius 20.

The correct answer is not among the other choices.

The correct answer is not among the other choices.

Explanation:

The figure below shows , which matches this description, along with its chord :

By way of the Isosceles Triangle Theorem,  can be proved equilateral, so .

This answer is not among the choices given.

### Example Question #3 : How To Find The Length Of A Chord

Give the length of the chord of a  central angle of a circle with radius .

Explanation:

The figure below shows , which matches this description, along with its chord  and triangle bisector

We will concentrate on , which is a 30-60-90 triangle. By the 30-60-90 Theorem,

and

is the midpoint of , so

### Example Question #4 : How To Find The Length Of A Chord

Figure NOT drawn to scale

In the figure above, evaluate .

Explanation:

If two chords of a circle intersect inside the circle, the product of the lengths of the parts of each chord is the same. In other words,

Solving for  - distribute:

Subtract  from both sides:

Divide both sides by 20:

### Example Question #5 : How To Find The Length Of A Chord

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

Evaluate .