# GMAT Math : Chords

## Example Questions

### Example Question #1 : Chords

The chord of a  central angle of a circle with area  has what length?

Explanation:

The radius  of a circle with area  can be found as follows:

The circle, the central angle, and the chord are shown below:

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

### Example Question #102 : Circles

The chord of a  central angle of a circle with area  has what length?

Explanation:

The radius  of a circle with area  can be found as follows:

The circle, the central angle, and the chord are shown below, along with , which bisects isosceles

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

and

The chord  has length twice this, or

### Example Question #1 : Calculating The Length Of A Chord

The chord of a  central angle of a circle with circumference  has what length?

Explanation:

A circle with circumference  has as its radius

.

The circle, the central angle, and the chord are shown below, along with , which bisects isosceles :

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

has half the length of , so

and

The chord  has length twice this, or

### Example Question #1 : Calculating The Length Of A Chord

The chord of a  central angle of a circle with area  has what length?

Explanation:

The radius  of a circle with area  can be found as follows:

The circle, the central angle, and the chord are shown below:

By way of the Isosceles Triangle Theorem,  can be proved a 45-45-90 triangle with legs of length . Its hypotenuse has length  times this, or

This is the correct response.

### Example Question #1 : Calculating The Length Of A Chord

The chord of a  central angle of a circle with circumference  has what length?

Explanation:

A circle with circumference  has as its radius

.

The circle, the central angle, and the chord are shown below:

By way of the Isosceles Triangle Theorem,  can be proved a 45-45-90 triangle with legs of length 30. By the 45-45-90 Theorem, its hypotenuse - the chord of the central angle - has length  times this, or . This is the correct response.

### Example Question #1 : Calculating The Length Of A Chord

Consider the Circle :

(Figure not drawn to scale.)

If is a  angle, what is the measure of segment ?

Explanation:

This is a triangle question in disguise. We have a ninety-degree triangle with two sides made up of the radii of the circle. This means the other two angles ( and ) must be  each.

Use the 45/45/90 triangle ratios to find the final side. Additionally, you could use Pythagorean Theorem to find the missing side.

45/45/90 side length ratios:

Segment

Or, using the Pythagorean Theorem,  by rearranging it and solving for , the hypotenuse, which in this case is segment :

### Example Question #107 : Circles

Calculate the length of a chord in a circle with a radius of , given that the perpendicular distance from the center to the chord is .

Explanation:

We are given the radius of the circle and the perpendicular distance from its center to the chord, which is all we need to calculate the length of the chord. Using the formula for chord length that involves these two quantities, we find the solution as follows, where  is the chord length,  is the perpendicular distance from the center of the circle to the chord, and  is the radius:

### Example Question #1 : Calculating The Length Of A Chord

The chord of a  central angle of a circle with circumference  has what length?

Explanation:

A circle with circumference  has as its radius

.

The circle, the central angle, and the chord are shown below:

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

### Example Question #1 : Calculating The Length Of A Chord

The arc  of a circle measures  and has length . Give the length of the chord .

Explanation:

The figure referenced is below.

The arc is  of the circle, so the circumference of the circle is

.

The radius is this circumference divided by , or

.

is, consequently, the hypotenuse of an isosceles right triangle with leg length ; by the 45-45-90 Triangle Theorem, its length is  times this, or