Tim's Solution

From: Tim, age 15, class: 9 geometry

Answer:

The answer to this problem is 81.

Explanation:

First you find the the sides of the two given squares which are 7
times 5 and 2 times four. From there you can find the area of the
other 2 boxes by multiplying 4 and 7,and 2 and 5.Then you add all
the products together 35, 28, 10, 8 which equals 81.




From: A GeoPOW Mentor

>The answer to this problem is 81.
>
>First you find the the sides of the two given squares which are 7
>times 5 and 2 times four. From there you can find the area of the
>other 2 boxes by multiplying 4 and 7,and 2 and 5.Then you add all
>the products together 35, 28, 10, 8 which equals 81.

Hi Tim.  Can you say some more about how you know those are the right numbers?
Why not choose some different factors of 35 and 8 (like 35 and 1, and 8 and 1,
maybe)?

-[the mentor]

--
[the mentor], for the Geometry Problem of the Week
http://forum.swarthmore.edu/geopow/ 

Brian's Solution

 
From: Brian, age 15, class: Geometry 

Answer:

The area of the square is 86.

Explanation:

I added 35 and 35 and got 70 and then I added 8 and 8 and I got 16
and then  I added and got 86.



From: A GeoPOW mentor

>The area of the square is 86.
>
>I added 35 and 35 and got 70 and then I added 8 and 8 and I got 16
>and then  I added and got 86.

Hi Brian.  It's a good idea to explain _why_ you did what you did.  The
numbers that you're given are areas, not edges.  Give it another try.

-[the mentor]

--
[the mentor], for the Geometry Problem of the Week
http://forum.swarthmore.edu/geopow/

Mary's Solution

From: Mary, age 15, class: 10

Answer:

The area of the square is 81.

Explanation:

I used quess and check.



From: [the mentor]

>The area of the square is 81.
>
>I used quess and check.

Hi Mary.  What did you guess?  How did you check it?  Pretend that you are
explaining it to a student who wasn't sure how to solve it.

-[the mentor]

--
[the mentor], for the Geometry Problem of the Week
http://forum.swarthmore.edu/geopow/

Konrad's Solution

From: Konrad, age 13, class: Geometry

Answer:

The area of the square is 77 sq units.

Explanation:

I got the factors of 35 = 1,5,7,35
and the factors of8  = 1,2,4,8

And worked it out getting 5,7  2,4

5*4 = 20
7*2 = 14

35+8+20+14=77



From: [the mentor]

>The area of the square is 77 sq units.
>
>I got the factors of 35 = 1,5,7,35
>and the factors of8  = 1,2,4,8
>
>And worked it out getting 5,7  2,4
>
>5*4 = 20
>7*2 = 14
>
>35+8+20+14=77

Hi Konrad.  You're missing an important part of the problem - that the
enclosing figure is a square.  See how that affects your answer.

-[the mentor]

--
[the mentor], for the Geometry Problem of the Week
http://forum.swarthmore.edu/geopow/

Michael's Solution

From: Michael, age 16, class: Advanced Geometry

Answer:

I concluded that the final area of the square is 81.

Explanation:

I first decided that to be a square both sides of the square had to be
equal. So I first set the part of the square that equaled 35 in area
to the top and bottom equaling 7, and the two sides equaling 5. I then
tried to think of two numbers that eight was divisible by to see if
they were able to add with the two numbers of the section of the
square that equals 35. I  came up with 4, for the two sides, and 2 for
the bottom and top. This meant that the distance from one side of the
square was 9 by 9. I then multiplied 9 and 9 together to get the whole
area of the square and this equaled 81, the solution. I found another
solution also. I set the smaller square in the bottom right corner to
1/2 on the sides, and 16 on the bottom and top. I then did the same
process and came up with 17 and 1/2 for the two sides of the upper
right section, and 2 for the bottom and top of the shape. This meant
that the distance across the whole side was 18. 18^2 equals 324, the
area of the other square, and the other solution.



From: [the mentor]

>I concluded that the final area of the square is 81.
>
>I first decided that to be a square both sides of the square had to be
>equal. So I first set the part of the square that equaled 35 in area
>to the top and bottom equaling 7, and the two sides equaling 5. I then
>tried to think of two numbers that eight was divisible by to see if
>they were able to add with the two numbers of the section of the
>square that equals 35. I  came up with 4, for the two sides, and 2 for
>the bottom and top. This meant that the distance from one side of the
>square was 9 by 9. I then multiplied 9 and 9 together to get the whole
>area of the square and this equaled 81, the solution. I found another
>solution also. I set the smaller square in the bottom right corner to
>1/2 on the sides, and 16 on the bottom and top. I then did the same
>process and came up with 17 and 1/2 for the two sides of the upper
>right section, and 2 for the bottom and top of the shape. This meant
>that the distance across the whole side was 18. 18^2 equals 324, the
>area of the other square, and the other solution.

Hey Michael.  Super job!  That's a very nice explanation.

-[the mentor]

--
[the mentor], for the Geometry Problem of the Week
http://forum.swarthmore.edu/geopow/

Sarah's Solution

From: Sarah, age 15, class: 10/Geometry

Answer:

The area of the whole square is 86^2.

Explanation:

First, I saw that the square was divided in 2 pairs of congruent
sections.  The section with 35^2 area was congruent to the section
below it.  And the section with 8^2 area was congruent to the section
above it.  I simply concluded that if you added the given areas with
the conjectured areas, they would add up to the total area of the
whole square, which was 86^2.



From: [the mentor]

>The area of the whole square is 86^2.
>
>First, I saw that the square was divided in 2 pairs of congruent
>sections.  The section with 35^2 area was congruent to the section
>below it.  And the section with 8^2 area was congruent to the section
>above it.  I simply concluded that if you added the given areas with
>the conjectured areas, they would add up to the total area of the
>whole square, which was 86^2.

Hi Sarah.  We don't know that it's split into two areas of each 8 and 35.  We
only know that the area of one region is 8 and the area of one region is 35.
We don't know much of anything about the other two regions.  Don't assume that
those lines split anything in half.  And you're ignoring an important part of
the problem - that the surrounding box is a square.  Give it another try.

 -[the mentor]

--
[the mentor], for the Geometry Problem of the Week
http://forum.swarthmore.edu/geopow/