### All ISEE Upper Level Math Resources

## Example Questions

### 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.

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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 : Chords

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

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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

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

**Possible Answers:**

**Correct answer:**

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

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

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

The correct answer is not among the other choices.

**Correct answer:**

The correct answer is not among the other choices.

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 .

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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:

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