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Five
Squares on a Rhombus
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Strategies
and Hints
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From the Pythagorean Theorem, the diagonal of a 6-by-6 square is
.
Half this length is 
.
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The key triangle is shaded in the drawing below.
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Since the two short sides of the shaded triangle are known, and its obtuse angle is 120° (45° + 30° + 45°), the Law of Cosines can be used to find the length of the third side.
Solution
The
shaded triangle is the key to the solution.
Each short side equals .
Using the Law of Cosines with an angle of 120°, the long side of the triangle
is 
.
So, the area of the square is 54 square units.
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