King Frostine Wages War

 

I'm very proud of this.  I wrote it  a couple years ago during finals.  Finals are boring!  I had only done one lesson of this 3-lesson unit.  I was getting ready to do the rest when we came back from break in January.
I made up the story, but not the names or 'races.'  Don't sue me.  I'm not making money off of this!

Geometry, "Center" of a Triangle, a Task from King Frostine

You live in the land of Sunderedcounter in Intermedio Tierra where King Frostine rules over all of the Lilliputians.  You are the king's head mathematician.

John Hurt's voice

 

There is some disarray in the land.  King Frostine is ready to wage war with three nearby cities Crystalgarth, Sunderfen, and Sharpetide.  His plan is to use catapults to continuously launch water balloons at each city until they surrender.

The assault will go fine as long as the liliputians have access to a camp nearby.

King Frostine only has enough resources to set up only one camp as a refilling station.

 
King Frostine must pick the location of his camp very carefully.  The camp must be exactly the same distance away from each Crystalgarth, Sunderfen, and Sharpetide.


The reasons for this are political.  King Frostine's biggest supporters, Lemel Hailglow, Hazel Flameshimmer, and Bracken Goldtree, each prefer he focus on a different city. 


Lemel wants to attack Crystalgarth.  The mermaids from Crystalgarth are always singing limericks about the liliputians right outside Lemel's house.

Hazel wants to attack Sunderfen.  The dragons of Sunderfen are always dropping jello-filled water balloons onto Hazel's daisies.

Bracken wants to attack Sharpetide.  The centaurs of Sharpetide are always stealilng Bracken's pumpkins.

King Frostine must pick a location that is equidistant to each so he isn't showing any favoritism.





He needs your help.  You have taken a map of Intermedio Tierra and connected Crystalgarth, Sunderfen, and Sharpetide.  This has created a triangle.

 



King Frostine set each of your apprentices Brelynd, Jarthan, Tolbain, and Camnar the task to determine where the camp should be located.


 

Brelynd thinks we should locate it at the centroid.
Jarthan thinks we should locate it at the orthocenter.
Tolbain thinks we should locate it at the incenter.
Camnar thinks we should locate it at the circumcenter.

Which place should we locate the camp of water balloon refills?

Use geometry vocabulary to explain this.


If you're not a math teacher, then you can stop reading here!



The following pages contain a map of Intermedio Tierra.

Calculate the distance from the camp to each of the cities.

Brelynd thinks we should put the camp at the centroid.  Use this graph to locate the location (ordered pair) of the centroid.  Be sure to label the midpoint of each side of the triangle.  Use a ruler to draw the medians.  Graph the centroid on the map.

Jarthan thinks we should put the camp at the orthocenter.  Use this graph to locate the location (ordered pair) of the orthocenter.  Use a ruler to make a (GOOD) sketch of each altitude.  Remember altitudes are perpendicular, and should be labeled as such.  Make a (GOOD) estimate of where the orthocenter is located. Label the orthocenter on the map.

Tolbain thinks we should put the camp at the incenter.  Use this graph to locate the location (ordered pair) of the incenter.  Use a compass and a straight-edge to construct each angle bisector.  Remember to label the angles as bisected.  Make a (GOOD) estimate of where the incenter is located. Label the incenter on the map.

Camnar thinks we should put the camp at the circumcenter.  Use this graph to locate the location (ordered pair) of the circumcenter.  Use a straight-edge to sketch each perpendicular bisector.  Remember to label the sides as bisected.  Determine where the circumcenter is located. Label the circumcenter on the map.

I put the same graph paper map on each page.
It's easiest if you use a right triangle.