Uniformity Area

Geometry Level pending

Two uniform parallelograms look like this:

The area of the blue-shaded area is...

NOTE: The shapes are not to scale in the image.

...greater than 1. ...greater than 2. ...equal to 1. ...equal to 2.

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1 solution

Azeez Daoud
Apr 21, 2018

We can start by calculating what K K is.

K = 7 3 = 4 K=7-3=4

K 3 = 4 3 = 1.333... 3 1.333... = 2.25 \frac{K}{3}=\frac{4}{3}=1.333...\ \\ \frac{3}{1.333...}=2.25

Now using the Pythagorean Theorem, we can determine the height of the smaller parallelogram.

a 2 + b 2 = c 2 a = 3 2 c = 2.25 1 2 + b 2 = 2.2 5 2 b 2 = 5.0625 1 b = 4.0625 A T r i a n g l e = b × 1 2 A T r i a n g l e = 2.0155... × 1 2 A T r i a n g l e = 1.00778... A T r i a n g l e > 1 a^{2}+b^{2}=c^{2} \\ a=3-2 \\ c=2.25 \\ 1^{2}+b^{2}=2.25^{2} \\ b^{2}=5.0625-1 \\ b=\sqrt{4.0625} \\ A_{Triangle}=\frac{b \times 1}{2} \\ A_{Triangle}=\frac{2.0155... \times 1}{2} \\ A_{Triangle}=1.00778... \\ A_{Triangle}>1

You could also say if b 2 > 4 b^{2}>4 then b > 2 b>2 and therefore the area will be greater than 1.

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