2022 AMC 12A Problems/Problem 3
Contents
Problem
Five rectangles, ,
,
,
, and
, are arranged in a square as shown below. These rectangles have dimensions
,
,
,
, and
, respectively. (The figure is not drawn to scale.) Which of the five rectangles is the shaded one in the middle?
Solution 2 (Area and Perimeter of Square)
The area of this square is equal to , and thus its side lengths are
. The sum of the dimensions of the rectangles are
. Thus, because the perimeter of the rectangle is
, the rectangle on the inside must have a perimeter of
. The only rectangle that works is
.
~mathboy100
Solution 2 (Perimeter of Square)
Note that the perimeter of the square is the sum of four pairs of dimensions (eight values are added). Adding up all dimensions gives us . We know that as the square's side length is an integer, the perimeter must be divisible by
. Testing out by subtracting all five pairs of dimensions from
, only
works since
, which corresponds with
.
~iluvme
Solution 3 (Observations)
Note that rectangle must be on the edge. Without loss of generality, let the top-left rectangle be
as shown below:
DIAGRAM READY SOON
We conclude that so we can determine Rectangle
Continuing with a similar process, we can determine Rectangles and
in this order. The answer is
as shown below.
DIAGRAM READY SOON
~MRENTHUSIASM
Solution 4 (Observations)
Let's label some points.
By finding the dimensions of the middle rectangle, we need to find the dimensions of the other 4 rectangles. By doing this, there is going to be a rule.
Rule:
Let's make a list of all the dimensions of the rectangles from the diagram. We have to fill in the dimensions from up above, but still apply to the rule.
By applying the rule, we get , and
.
By substitution, we get this list
This also tells us that the diagram is not drawn to scale, lol.
Notice how the only dimension not used in the list was and that corresponds with B so the answer is,
~ghfhgvghj10 & Education, the study of everything.
See Also
2022 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 2 |
Followed by Problem 4 |
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All AMC 12 Problems and Solutions |