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705-805 (Hard)|   Geometry|      
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dave13
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A rectangle is inscribed in a hexagon that has all sides of equal leng 25 Mar 2022, 03:53
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KarishmaB
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Bunuel

A rectangle is inscribed in a hexagon that has all sides of equal length and all angles of equal measure, as shown in the figure above. If the total area of the hexagon is x, then the sum of the areas of the two shaded regions is


A. x/6

B. x/4

C. x/3

D. x/2

E. 2x/3


PS20418


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Attachment:
1.png


KarishmaB
how about an alternative solution where you subtract an area of rectangle from an area of hexagon

i did this but smth went wrong :lol: \(\frac{3*x^2 \sqrt{3} }{2 }- 2x^2\sqrt{3}\)


area of rectangle i got this way: each shaded region represents isosceles triangle, so if i draw perpendicular line i get 30:60:90 trangle


hence length of rectangle is \(2x\sqrt{3}\) and width x

I would much rather use the symmetry of a regular hexagon and the triangles I can carve out of it as done above.
But if you wanted to use geometry as suggested by you, note that x is the area of the hexagon, not a side.

Since a hexagon is made up of 6 equilateral triangles, its area will be \(6 * \frac{\sqrt{3} * a^2}{4} = x\) ... (I)

If side of the hexagon is a, in the 30-60-90 shaded triangle (half of each shaded triangle), the side opposite to 60 is \(\frac{\sqrt{3}a}{2}\) which means the width of the rectangle is \(\sqrt{3}a\).

Area of rectangle is \(a*\sqrt{3}a = \sqrt{3}a^2 = \frac{2x}{3}\) (from (I) above)

Shaded area\( = x - \frac{2x}{3} = \frac{x}{3}\)
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KarishmaB thanks! btw what is the key property of pentagon to remember ? similar to the the symmetry of hexagon .... that it can be divided into 6 equilateral triangles
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A rectangle is inscribed in a hexagon that has all sides of equal leng Updated on: 08 Aug 2023, 04:03
Originally posted by KarishmaB on 28 Mar 2022, 06:41.
Last edited by KarishmaB on 08 Aug 2023, 04:03, edited 1 time in total.
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KarishmaB thanks! btw what is the key property of pentagon to remember ? similar to the the symmetry of hexagon .... that it can be divided into 6 equilateral triangles
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dave13
No one property of a figure will help you solve all questions. The key is to understand the figure really well.

Try out these posts on regular and irregular polygons:
https://anaprep.com/geometry-diagonals- ... d-polygon/
https://anaprep.com/geometry-question-o ... or-angles/
https://anaprep.com/geometry-question-o ... or-angles/
https://anaprep.com/geometry-a-tough-ir ... -question/

And here is a video discussing interior and exterior angles of polygons:
https://www.youtube.com/watch?v=qdhrxsf7gQY
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A rectangle is inscribed in a hexagon that has all sides of equal leng 01 Dec 2024, 05:45
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