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Sub 505 (Easy)|   Geometry|      
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Bunuel
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Quadrilateral ABCD is composed of four 30–60–90 triangles. If \(BD=10\ 26 Mar 2021, 05:00
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93% (01:52) correct 7% (03:10) wrong based on 44 sessions

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Quadrilateral ABCD is composed of four 30–60–90 triangles. If \(BD=10\sqrt{3}\), what is the perimeter of ABCD?

A. 38
B. 40
C. 42
D. 46
E. 47

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Quadrilateral ABCD is composed of four 30–60–90 triangles. If \(BD=10\ 26 Mar 2021, 05:06
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Bunuel

Quadrilateral ABCD is composed of four 30–60–90 triangles. If \(BD=10\sqrt{3}\), what is the perimeter of ABCD?

A. 38
B. 40
C. 42
D. 46
E. 47

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It makes ∆ABC and BCD both equilateral triangles

Now, BD = 2*Height of one equilateral triagle \(= 2*\frac{√3}{2}*Side = 10√3\)

i.e. \(Side = 10\)

\(Perimeter = 4*10 = 40\)



Answer: Option B
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Quadrilateral ABCD is composed of four 30–60–90 triangles. If \(BD=10\ 26 Mar 2021, 05:29
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let us consider two diagonals intersect at point O; as we know in 30,60, 90 degree triangle sides are in the ratio 1:sqrt3:2

using triangle OAD (30,60 and 90 degree angle triangle) we know because of symmetry OD=5*sqrt3, so AD=5*sqrt3*2/Sqrt3=10

Perimeter=4*AD=4*10=40
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Quadrilateral ABCD is composed of four 30–60–90 triangles. If \(BD=10\ 18 Apr 2021, 03:58
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sumitkrocks
let us consider two diagonals intersect at point O; as we know in 30,60, 90 degree triangle sides are in the ratio 1:sqrt3:2

using triangle OAD (30,60 and 90 degree angle triangle) we know because of symmetry OD=5*sqrt3, so AD=5*sqrt3*2/Sqrt3=10

Perimeter=4*AD=4*10=40
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How do you prove that OD = half of BD?
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RobertoFerro
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Quadrilateral ABCD is composed of four 30–60–90 triangles. If \(BD=10\ 20 Apr 2021, 14:43
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- From the info we know the BD is 10\sqrt{3}. This means from D to the center of the Kite the line's length is 5\sqrt{3}.
- We see angle D is bisected, therefore we have it cut in half. i.e. 30.
- We see from the properties of a 30-60-90 triangle, that the opposite of the 60 degree angle has a length of x\sqrt{3}. From above this means x=5
- Since x = 5, then the hypotenuse is 5*2 = 10.
- If once side is 10m the. 4 sides is 40.

Answer = B

(I wanted to upload images to better explain but GMAT club wouldn't let me. Hope this helps.
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