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Ratio of Areas in a Square
Given: a square
with side length
and diagonal
. Place point
on
and let
. Connect
to divide the square into four areas. Given
, calculate the ratio of areas. For example, let
.
Contributed by:
Abraham Gadalla
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The key here is that the height of
equals the height of
, so the ratio of the areas between those two triangles is
.
Hence,
.
Similarly, comparing areas,
. Therefore |
.
Substituting gives
.
At this point, one can calculate the height
of
as
.
Then
and
.
Also,
.
After simplification, the ratio of the four areas is:
.
RELATED LINKS
Ratio
(
Wolfram
MathWorld
)
PERMANENT CITATION
Abraham Gadalla
"
Ratio of Areas in a Square
"
http://demonstrations.wolfram.com/RatioOfAreasInASquare/
Wolfram Demonstrations Project
Published: August 15, 2011
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