How to find the length of a chord - ISEE Upper Level Quantitative Reasoning

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 =

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

Chord length =

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.

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 .

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 .

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

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

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:

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:

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:

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:

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

Substituting:

Distributing, then solving for :

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.

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

Chord length =

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 =

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

Chord length =

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.

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 .

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 .