## Problem: Two
Squares (click here to see the problem statement)

My Solutions:

#### GRAPHIC SOLUTION.

#### We can see that when AP is relatively small, the combined
area of the squares formed by AP and PB is slightly less than
the area of the square formed by AB:

**The combined area decreases as we increase the length AP:**

**When P is at the midpoint of AB, the combined areas
formed by the squares AP and PB equal one-half the area of the
square formed by AB.**

**As AP is increased beyond 1/2 the length of AB, the combined
area will now increase, approaching the area of the square formed
by AB:**

**The MAXIMUM combined area is when P is at either A or B:
In these cases, one the othe "squares" has an area of
zero and the other is equal to the area of the square formed by
AB.**

**Point P should be located at the midpoint of AB to MINIMIZE
the combined areas of the squares formed by AP and PB.**

**For
an ANIMATED VERSION of the graphic representation,**

Open Geometer's Sketchpad then Click Here

**ALGEBRAIC SOLUTION.**

**Let's look at AP in increments of 1/8. When AP = 1/8AB,
then PB = 7/8AB. The combined areas of the squares formed by each
segment will then be**

**When AP = 2/8AB, then PB = 6/8AB, so the combined area will
be**

**When AP = 3/8AB, then PB = 5/8AB, so the combined area will
be**

**When AP = 4/8AB, then PB = 4/8AB, so the combined area will
be**

**When AP = 5/8AB, then PB = 3/8AB, and the combined area
will be the same as AP = 3/8AB and PB = 5/8AB. Again, the minimum
combined area is 1/2 of the area of the square formed by AB.**