Jill Avery

1

• DUDENEY DISSECTION

• By
• Jill Avery

2
An Equilateral Triangle

• Each side is 10 cm long.

A P Q
B
C

• Find the midpoints of the two legs, so that each
  segment equals 5 cm.

• Connect the midpoints

3
The Dissection

• Construct QR such that
• QR?25?3cm
• QR?6.85cm

A P Q
Y X B

C R

• Construct RS5cm (BRgtSC)

• Construct the perpendiculars to RQ from points P
  and S.

S
4
The Proof

• PQ 5cm because ?ABC is equilateral and P and Q
  are the midpoints.
• The ?QRC
• By the Law of Sines
• sin ?QRC sin?QCR
• QC QR
• sin ?QRC ?3/2_
• 5 5 ??3
• Therefore,
• sin ?QRC ??3
• 2

A 5 5 P 5
Q Y 5
5
X 6.58 B
C R
5 S
5
The Proof

• The ? RSY
• sin ?YRS YS/ RS
• know sin ?YRS and RS
• Therefore,
• YS 5 ??3
• 2
• We can see by the reassembled pieces that one
  side of the rectangle is 2YS 5 ??3

A 5 5 P 5
Q Y 5
5
X 6.58 B
C R
5 S
6
The Proof

• The area of the whole triangle is 25?3
• s? (30) 15
• A ? 15(15-10)(15-10)(15-10)
• 25 ?3
• This must also be the area of the rectangle
• We must find the length of the other side of the
  rectangle by dividing the area by the length of
  the side we have
• 25 ?3 5 ??3
• 5 ??3

A 5 5 P 5
Q Y 5
5
X 6.58 B
C R
5 S
7
The Proof

• Since the lengths of both sides came out to be
  the same, we have a square.

A 5 5 P 5
Q Y 5
5
X 6.58 B
C R
5 S
8
Separating the Pieces
A P Q
Y X B

C R S

• The triangle is cut so on the solid white lines,
  forming 4 pieces.

9
Reassembling of the pieces

• When reassembled as a square, the sides with like
  colors match against one another.

• R
• A(B,C)
• S
• P
• Q

• Each colored line is 5 cm long.