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25 Mar 2018, 09:05
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dave13
dave13 :
i have one awesome question 
niks18 , apparently able
to get a word in edgewise
addressed what to do
with a "2" that is now "1."
I'm hoping a third ratio is also clear, see below.
That ratio also must be divided by 2.
sadikabid27 , maybe this discussion or this post will help.
See below for a link to 30-60-90 triangle overview.
dave13 , yes, we create two identical 30-60-90 triangles at vertex U.
So let's take niks18 's answer and line up the ratios.
Normal = \(30: 60 : 90\)
Normal = \(1 : \sqrt{3} : 2\)
Normal = \(x : x\sqrt{3} : 2x\)
But we can't call long sides PU and UT "2x" or "2."
Both are hexagon sides.
Hexagon side length is not \(2x\) or \(2\)
It is \(x\). Or, if you're thinking in numeric multiples, \(1\)
2x or 2 got divided by 2.
What you do to one part of a ratio,
you must do to all parts.
Divide all ratio parts by 2
\(30 : 60 : 90\)
Adjusted = \(\frac{1}{2} : \frac{\sqrt{3}}{2} : 1\)
Adjusted = \(\frac{x}{2} : \frac{x\sqrt{3}}{2} : x\)
Length of ONE side of the big triangle =
\(\frac{x\sqrt{3}}{2} + \frac{x\sqrt{3}}{2} = x\sqrt{3}\)
Hope that helps.
P.S. For a quick review of 30-60-90 triangles as well as an interesting proof see HERE.
dave13 - last thing . . . one part of the answer you quoted may have confused you.
I wrote Regular hexagon, 120 degrees at each vertex, equilateral triangle
Highlight refers to the big triangle in the center, not the smaller right triangles we create.
BlacknYellow
GMAT 2: 700 Q48 V38
Posts: 12
15 Jan 2019, 16:29
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AbdurRakib
A hexagon is made up of 6 equilateral triangles (see drawing attached).
Each side of triangle PRT corresponds to 2* (The height of each equilateral triangle), so the perimeter will be 6*(height of each equilateral triangle) = (6\(x\sqrt{3}\))/2 or D
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hex.png [ 29.91 KiB | Viewed 5799 times ]
14 Aug 2019, 22:37
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AbdurRakib
An Engineer's method~
sinA/A=sinB/B=sinC/C
let the side of the isosc. triangle PQR be "y",
So, sin30/x = sin120/y
or, y = (3)^0.5 * x [since, sin30=1/2 & sin120=(3)^0.5/2]
Hence, perimeter = 3*y = 3*x*(3)^0.5 [since, triangle PRT is equilat.]
OA=D
