An equilateral triangle of perimeter 84 has sidelength one-third of this, or 28. 

Construct this triangle and its circumscribed circle, as well as a perpendicular bisector to one side and a radius to one of that side's endpoints:

Each side of the triangle has measure 28, so . Also, the triangle formed by the segments, by symmetry, is a 30-60-90 triangle. Therefore, by the 30-60-90 Theorem,

and .

This is the radius, so the circumference is  times this, or 

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 is the radius, so the circumference is

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

By way of the Isosceles Triangle Theorem,  can be proved a 45-45-90 triangle with hypotenuse 40. By the 45-45-90 Theorem, its legs, both radii, have length that can be determined by dividing this by , so

This is the radius, so the circumference is

If a right triangle is inscribed inside a circle, then the arc intercepted by the right angle is a semicircle, making the hypotenuse of triangle a diameter. 

The length of a hypotenuse of a 30-60-90 triangle is twice that of its short leg, so the hypotenuse of this triangle will be twice , or

.

This is the diameter, also, so the circumference is  times this, or .

Write the formula to find the circumference of a circle.

Substitute the diameter into the formula.

Write the formula for the area of the circle.  The radius is needed to find the circumference of the circle.

Substitute the area.

The radius of the circle is 3.  Substitute this in the circumference formula.

The correct answer is: 

The circumference of a circle with radius 500 feet is

 feet.

Rafael runs this distance twice, then he runs from Point C to D, which is about one-fifth of this distance. Therefore, Rafael's run will be about

 feet.

Divide by 5,280 to convert to miles:

 miles.

Of the given choices,  miles is the closest to the correct length.

First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 500 feet; this length is

 feet.

The distance around the track is about

 feet.

Mike runs around the track five times, which is a distance of about

 feet.

He then proceeds to Point B, which is an additional 500 feet, and to Point C, which is half the semicircle, or

 feet.

Therefore, Mike runs about

 feet.

Divide by 5,280 to convert to miles:

 miles.

Of the choices, the closest is  miles.

First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 400 feet; this length is

 feet.

The distance around the track is about

 feet.

Divide this into 5,280 feet (one mile):

This means that Jennifer will run two times around the track to return to Point A. She will have 1,624 feet to go, so she will do the following:

She will run 400 feet from A to B, leaving  feet;

She will run 628 feet from B to D, leaving  feet;

She will run 400 feet from D to E, leaving  feet;

She will then run the remaining 196 feet from Point E toward Point A.

The correct response is that she will be between Point E and Point A.

A circle of radius 600 feet will have a circumference of 

 feet.

Boris will run one mile, or 5,280 feet, which will be about

 tiimes the circumference of the track, or, equivalently, once around the track, plus another two-fifths of the circle. This means he will end up closest to point C.