# ISEE Upper Level Math : Circles

## Example Questions

### Example Question #5 : 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 #8 : 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 #9 : 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 #10 : How To Find The Length Of A Chord

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

Evaluate .

Explanation:

If a secant segment and a tangent segment are constructed to a circle from a point outside it, the square of the distance to the circle along the tangent is equal to the product of the distances to the two points on the circle along the secant; in other words,

Solving for :

Simplifying the radical using the Product of Radicals Principle, and noting that 36 is the greatest perfect square factor of 360:

### Example Question #11 : How To Find The Length Of A Chord

Figure NOT drawn to scale

In the above diagram, 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 :

Simplifying the radical using the Product of Radicals Principle, and noting that 25 is the greatest perfect square factor of 50:

### Example Question #12 : How To Find The Length Of A Chord

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

Evaluate .

Explanation:

If a secant segment and a tangent segment are constructed to a circle from a point outside it, the square of the distance to the circle along the tangent is equal to the product of the distances to the two points on the circle along the secant; in other words,

,

and, substituting,

Distributing and writing in standard quadratic polynomial form,

We can factor the polynomial by looking for two integers with product  and sum 24; through some trial and error, we find that these numbers are 32 and , so we can write this as

By the Zero Product Principle,

, in which case  - impossible since  is a (positive) distance; or,

, in which case  - the correct choice.

### Example Question #13 : How To Find The Length Of A Chord

Figure NOT drawn to scale

In the above diagram, 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 :

Simplifying the radical using the Product of Radicals Principle, and noting that the greatest perfect square factor of 96 is 16:

### Example Question #14 : How To Find The Length Of A Chord

Figure NOT drawn to scale

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

Evaluate .