2020 AMC 10B Problems/Problem 21
Contents
Problem
In square , points
and
lie on
and
, respectively, so that
Points
and
lie on
and
, respectively, and points
and
lie on
so that
and
. See the figure below. Triangle
, quadrilateral
, quadrilateral
, and pentagon
each has area
What is
?
Solution
Since the total area is , the side length of square
is
. We see that since triangle
is a right isosceles triangle with area 1, we can determine sides
and
both to be
. Now, consider extending
and
until they intersect. Let the point of intersection be
. We note that
is also a right isosceles triangle with side
and find it's area to be
. Now, we notice that
is also a right isosceles triangle and find it's area to be
. This is also equal to
or
. Since we are looking for
, we want two times this. That gives
.~TLiu
Solution 2
Since this is a geometry problem involving sides, and we know that is
, we can use our ruler and find the ratio between
and
. Measuring(on the booklet), we get that
is about
inches and
is about
inches. Thus, we can then multiply the length of
by the ratio of
of which we then get
We take the square of that and get
and the closest answer to that is
. ~Celloboy (Note that this is just a strategy I happened to use that worked. Do not press your luck with this strategy, for it was a lucky guess)
See Also
2020 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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All AMC 10 Problems and Solutions |
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