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## Example Questions

### Example Question #1 : Hexagons

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?

**Possible Answers:**

Point E

Point D

Point C

Point B

Point F

**Correct answer:**

Point E

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 #2 : Hexagons

A circle with circumference is inscribed in a regular hexagon. Give the perimeter of the hexagon.

**Possible Answers:**

None of these

**Correct answer:**

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

.

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