Expert: Paul Klarreich - 5/16/2007

Question
Perhaps youve heard of the famous English pirate named Blackbeard, but did
you ever hear about his Spanish cousing, Barbanegra? Probably not. he was
the mathematician of the high seas. When he buried his treasure, he did so in
a very algebraic manner. Really!

For example, once after a very successful raid on a rich merchant ship, he
took three of his most trusted henchmen onto the island with the loot to bury
it in a very special way. barbanegra stood by a certain palm tree with a long
string in his hand, Holding one end in his hand, he gave the other end to one
of his men and directed him to walk 6 varas straight south, then 8 varas due
east.

He gave another string to each of the other two men, instructing one to walk
a spot 6 varas due east and 2 varas due north of the tree. He told the third
man to walk to a spot that was 6 varas due south and 2 varas due west.


Once everyone was in position, the strings were stretched taut, forming two
straight lines. The spot where they crossed was the place that Barbanegra
wanted to dig a hole to store his treasure. And they did just that.

Your task now is to determine the coordinates of that "X" spot, given that
coordinate pair for the tree is (3, 4).

One piece of advice: isomg graph paper will help give you the general idea
forthe answer, but it will not give you the exact coordinates. They must be
expressed in mixed number form, which means no decimals either. So you
must use another of the techniques couvered in your algebra textbooks,
okay?

P.S. A vara is a legitimate unit of linear measure in the Spanish world.  

Answer
Questioner:   Star
Category:  Advanced Math
 
Subject:  Math precalculus
Question:  Perhaps you've heard of the famous English pirate named Blackbeard, but did you ever hear about his Spanish cousin, Barbanegra?

Probably not. he was the mathematician of the high seas. When he buried his treasure, he did so in a very algebraic manner. Really!

For example, once after a very successful raid on a rich merchant ship, he took three of his most trusted henchmen

>> Who we will call A,B,C.

onto the island with the loot to bury it in a very special way. barbanegra stood by a certain palm tree

>> which we will designate as D, and tentatively make  (0,0)

with a long string in his hand, Holding one end in his hand, he gave the

other end to one of his men

>> A

and directed him to walk 6 varas straight south, then 8 varas due east.

>> arriving at  A(8,-6)

He gave another string to each of the other two men, instructing one

>> B

to walk to a spot 6 varas due east and 2 varas due north of the tree.

>> Arriving at B(6,2)

He told the third man

>> C

to walk to a spot that was 6 varas due south and 2 varas due west.

>> Arriving at C(-2,-6)

Once everyone was in position, the strings were stretched taut, forming two straight lines.

>> designated AD and BC.

The spot where they crossed was the place that Barbanegra
wanted to dig a hole to store his treasure. And they did just that.

Your task now is to determine the coordinates of that "X" spot, given that coordinate pair for the tree is (3, 4).

>> OK, we'll fix that later.  Right now it's (0,0).

One piece of advice: isomg graph paper will help give you the general idea for the answer, but it will not give you the exact coordinates. They must be expressed in mixed number form, which means no decimals either. So you must use another of the techniques couvered in your algebra textbooks, okay?

P.S. A vara is a legitimate unit of linear measure in the Spanish world.

>> Sure.

Well, we have two lines AD and BC.  If we can derive equations for each, we will have two simultaneous equations to solve, and the solution will be the
coordinates of the point X.
........................
AD passes through (0,0) and (8,-6).  It has slope = -6/8 = -3/4, and it passes through (0,0) so its equation is:

y = -3/4 x  or  4y + 3x = 0
.......................
BC passes through  (6,2) and (-2,-6).  It has slope:
   2 - (-6)   2 + 6
m = -------- = ----- = 1
   6 - (-2)   6 + 2

Use the point-slope form:  y - y0 = m(x - x0)

y - 2 = 1(x - 6)
y - 2 = x - 6
x - y = 4
......................
Here are our two equations:

3x + 4y =  0   (I)
 x -  y =  4   (II)
4x - 4y = 16   (4II)
---------------------
7x = 16  (sum)
x = 16/7

16/7 - y = 4
16 - 7y = 28
  - 7y = 12
     y = -12/7

The coordinates, based on D = (0,0), are  (16/7, -12/7)

But we have to translate to  D = (3,4), so we have to add (3,4) to those coordinates:

(16/7 + 3, -12/7 + 4) =
(16/7 + 21/7, -12/7 + 28/7) =
(37/7, 16/7)

which you can change to mixed numbers, if you like.