## Example Questions

### Example Question #13 : Geometry Archimedes High School has an unusual track in that it is shaped like a regular hexagon, as above. Each side of the hexagon measures 264 feet.

Alvin runs at a steady speed of seven miles an hour for twelve minutes, starting at point A and working his way clockwise. When he is finished, which of the following points is he closest to?

Point C

Point B

Point D

Point F

Point E

Point E

Explanation:

Alvin runs at a rate of seven miles an hour for twelve minutes, or hours. The distance he runs is equal to his rate multiplied by his time, so, setting in this formula:  miles.

One mile comprises 5,280 feet, so this is equal to feet

Since each side of the track measures 264 feet, this means that Alvin runs sidelengths. ,

which means that Alvin runs around the track four complete times, plus four more sides of the track. Alvin stops when he is at Point E.

### Example Question #14 : Geometry

A circle with circumference is inscribed in a regular hexagon. Give the perimeter of the hexagon.   None of these  Explanation:

Below is the figure referenced; note that the hexagon is divided by its diameters, and that an apothem—a perpendicular bisector from the center to one side—has been drawn. The circle has circumference ; its radius, which coincides with the apothem of the hexagon, is the circumference divided by : The hexagon is divided into six equilateral triangles. One, , is divided by an apothem of the hexagon - a radius of the circle - into two 30-60-90 triangles, one of which is . Since has length 30, and it is a long leg of , then short leg has length  is the midpoint of , one of the six congruent sides of the hexagon, so ;

this makes the perimeter of the hexagon six times this, or . 