A Question of Watering
A farmer has a plot of land in the shape of a square that is 100m on a side. An irrigation system can be installed with the option of one large circular sprinkler or nine small sprinklers, as shown below: Please help farmer Euclid decide which plan will provide water to the greatest percentage of land in the field. (At this point we will not be considering cost of the systems and/or water usage).

these 9 circles should be tangent to each other.

Roping a Square
Given a square of any size, stretch a rope tightly around it. Now take the rope off, add 100ft to it and put the extended rope back around the square so that the new rope makes a square around the original square. Make sure there is uniform spacing between the original square and the square formed by the rope. What is the distance between a side of the original square and the corresponding side of the rope square?

Relationships

Part A. Use a dynamic geometry software to measure the circumference (C) and the diameter (d) of several circles with different radii. Look at the ratios formed by C/d. What do you notice? Do you recognize this number?

Part B. What happens to the circumference of a circle if you double the diameter? If you triple the diameter? If you halve the diameter? As the diameter increases (or decreases) in measure, how does the circumference change? Why does this change occur?

Circles and Angles
The vertex of an angle can appear on, inside, or outside a circle. How does the location and measure of the vertex angle compare with the measure(s) of the arc(s) it intercepts?

Central Angle	Inscribed Angle
Secant/Secant Angle	Tangent/Tangent Angle

Tangent Lines
Using patty paper, construct a segment that is tangent to a circle at a given point. Explain this construction.

Chords of a Circle
Using patty paper, create two segments that have endpoints on a given circle. Move the segments until they cross each other and form four small segments.

Find a relationship between two of the small segments and the other two small segments.

Dartboard Dilemma
Since Miss Sheehy is an Olympic archer and a mathematician, she is fascinated by the geometry of a dartboard/target. She is currently investigating how the area of the outside shaded region compares with the area of the inside shaded region. How do they compare?

Buried Treasure
Mean Gene and his pirate cohorts buried a treasure on the island depicted below. The treasure is buried at a point X which is equidistant from Bob's and Pearl's place and such that m<XPB is 30 degrees. Using only a compass and a straightedge, locate the treasure on the map and mark it with an X.