# Thread: [SOLVED] Geometry Treasure Pirate Problem

1. ## [SOLVED] Geometry Treasure Pirate Problem

My teacher assigned this very confusing, at least to me, problem.

(Summed up by me.): Some people found a treasure map, authenticated it and now looking for it. Here's what it says:

"Start from the tree, walk directly to the falcon-shaped rock. Count your paces as you walk. Turn a quarter-circle to the right and walk the same number of paces. When you reach the end put a stick in the ground. Return to the tree and likewise walk to the owl-shaped rock, again counting your paces. Turn a quarter-circle to the left, walk the same number of paces, and put another stick in the ground. Connec the two sticks with a rope. The treasure is beneath the midpoint of the rope."

HOWEVER, the tree was no longer on the island. All that was left was the two weird shaped rocks. The people gave up looking for the treasure.

Did the people give up too soon? Is it possible to find the treasure without the tree? Explain and give a rough sketch.

My answer is no. I think there are way to many possiblities. I drew out a lot of sketches and my head starts to hurt. So, if you know of the answers would you please explain it to me? Thanks

2. As it is summed up by you, or as posted, I am seeing millions of possible answers.

Can you write your teacher's question exactly as she/he spoke it to you? Maybe we can get a different summary, and a unique solution too.

This portion:
"Start from the tree, walk directly to the falcon-shaped rock. Count your paces as you walk. Turn a quarter-circle to the right and walk the same number of paces. When you reach the end put a stick in the ground. Return to the tree and likewise walk to the owl-shaped rock, again counting your paces. Turn a quarter-circle to the left, walk the same number of paces, and put another stick in the ground. Connec the two sticks with a rope. The treasure is beneath the midpoint of the rope."

3. Hello, John!

Some people found a treasure map, authenticated it and now looking for it.
Here's what it says:

"Start from the tree, walk directly to the falcon-shaped rock.
Count your paces as you walk.
Turn a quarter-circle to the right and walk the same number of paces.
When you reach the end put a stick in the ground.

Turn a quarter-circle to the left, walk the same number of paces,
and put another stick in the ground. Connect the two sticks with a rope.
The treasure is beneath the midpoint of the rope."

HOWEVER, the tree was no longer on the island.
All that was left was the two weird-shaped rocks.
The people gave up looking for the treasure.

Did the people give up too soon?
Is it possible to find the treasure without the tree? . . . . Yes!

This is a classic problem.
Code:
Q
X    o
P      o    |*
o      |    | *
*|      |    |  *
* |      |    |   *
*  |      |    |    *
*   |      |    |     *
*    |      |    |      *
*     |    B |    |       *
F o------*----*-*----*--------o O
*   A    | C    D    *
*     |       *
*  |   *
o
T

We start at $\displaystyle T$, walk to $\displaystyle F$, turn 90° to the right,
. . and walk to point $\displaystyle P$, where $\displaystyle FP = TF$.

We start at $\displaystyle T$, walk to $\displaystyle O$, turn 90° to the left,
. . and walk to point $\displaystyle Q$, where $\displaystyle OQ = TO$.

The treasure is buried at $\displaystyle X$, the midpoint of $\displaystyle PQ$.

Draw perpendiculars $\displaystyle PA,\:TB,\:XC,\:QD$ to $\displaystyle FO$.

Since $\displaystyle X$ is the midpoint of $\displaystyle PQ$, $\displaystyle XC$ is the average of $\displaystyle PA$ and $\displaystyle QD$.
. . That is: .$\displaystyle XC \:=\:\frac{PA + QD}{2}$ .[1]

We find that: .$\displaystyle \Delta PAF \cong \Delta FBT\quad\Rightarrow\quad PA = FB$ .[2]

We find that: .$\displaystyle \Delta QDO \cong \Delta OBT\quad\Rightarrow\quad QD = BO$ .[3]

Substitute [2] and [3] into [1]: .$\displaystyle XC\:=\:\frac{FB + BO}{2} \:=\:\frac{1}{2}FO$

Hence, the "height" of $\displaystyle X$ is half of the distance between the two rocks.

Since $\displaystyle X$ is the midpoint of $\displaystyle PQ$, it follows that: $\displaystyle C$ is the midpoint of $\displaystyle AD$.

From the congruent right triangles: .$\displaystyle FA = TB$ and $\displaystyle DO = TB$
. . That is, the two end segments of $\displaystyle FO$ are equal.
Hence, $\displaystyle C$ is the midpoint of $\displaystyle FO$.

To locate the treasure $\displaystyle X$, we don't need the tree.

Locate $\displaystyle C$, the midpoint of $\displaystyle FO$.
From $\displaystyle C$, walk perpendicular to $\displaystyle FO$ a distance of $\displaystyle \frac{1}{2}FO$
. . and there is the treasure!