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The figure shown is a rhombus in which the measure of angle A = 120°.

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The figure shown is a rhombus in which the measure of angle A = 120°.  [#permalink]

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New post 20 Mar 2018, 22:23
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The figure shown is a rhombus in which the measure of angle A = 120°. What is the ratio of the length of AC to the length of DB?


A. \(1:2\sqrt{3}\)

B. \(1:2\sqrt{2}\)

C. \(1:2\)

D. \(1:\sqrt{3}\)

E. \(1:\sqrt{2}\)


Attachment:
2018-03-21_1019.png
2018-03-21_1019.png [ 7.44 KiB | Viewed 1580 times ]

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Re: The figure shown is a rhombus in which the measure of angle A = 120°.  [#permalink]

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New post 20 Mar 2018, 22:24
Bunuel wrote:
Image

The figure shown is a rhombus in which the measure of angle A = 120°. What is the ratio of the length of AC to the length of DB?


A. \(1:2\sqrt{3}\)

B. \(1:2\sqrt{2}\)

C. \(1:2\)

D. \(1:\sqrt{3}\)

E. \(1:\sqrt{2}\)


Attachment:
2018-03-21_1019.png



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Re: The figure shown is a rhombus in which the measure of angle A = 120°.  [#permalink]

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New post 20 Mar 2018, 23:02
Bunuel wrote:
Image

The figure shown is a rhombus in which the measure of angle A = 120°. What is the ratio of the length of AC to the length of DB?


A. \(1:2\sqrt{3}\)

B. \(1:2\sqrt{2}\)

C. \(1:2\)

D. \(1:\sqrt{3}\)

E. \(1:\sqrt{2}\)


Attachment:
The attachment 2018-03-21_1019.png is no longer available


IMO D.

quadrilateral ABCD is a rhombus... all the sides are equal in length and opposite sides are parallel to each other.

Let the length of side of rhombus be a. So the triangle formed by joining AC is equilateral as two of its sides are equal ( sides of rhombus ) and one angle is 60 degrees. ( opposite angles of a rhombus are supplementary)

AC = a.

Drop a line perpendicular to line AB from point D and let it intersect the line AB extended out at E.

Triangle ADE is 30-60-90 triangle.

So AE is a / 2... sine rule. as the hypotenuse of this 30-60-90 triangle is AD (a)

Hence the Side BE is a + a/2 = 3a/2

This is the side of a right angled triangle DEB...
and by pythagoras theorem -> DE^2 + BE^2 = DB^2

so \(DB^2 = (\sqrt{3} a/2)^2 + (3a/2)^2\)
\(DB = \sqrt{3}a\)

Hence AC : DB = a : \sqrt{3}a

AC : DB = 1 : \sqrt{3}

Option D.

Attached is the image to explain the diagrams & calculations...

Best,
Gladi
Attachments

rhombus.jpg
rhombus.jpg [ 1.04 MiB | Viewed 1337 times ]

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Re: The figure shown is a rhombus in which the measure of angle A = 120°.  [#permalink]

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New post 24 Mar 2018, 16:15
1
Bunuel wrote:

The figure shown is a rhombus in which the measure of angle A = 120°. What is the ratio of the length of AC to the length of DB?

A. \(1:2\sqrt{3}\)

B. \(1:2\sqrt{2}\)

C. \(1:2\)

D. \(1:\sqrt{3}\)

E. \(1:\sqrt{2}\)

Attachment:
2018-03-21_1019xxxx.png
2018-03-21_1019xxxx.png [ 10.14 KiB | Viewed 1135 times ]

Geometry plus clues . . .

THREE clues suggest that a (30-60-90)
triangle is likely to help solve:

• The figure is a rhombus
Equal side lengths, diagonals bisect vertices,
diagonals are perpendicular
(create 90° angles)
,
opposite angles are equal

• Vertex A = 120° (= C)
A and C's bisected angle = 60° + 60°
From above, diagonals create 90° angles
180° in a triangle: 60 . . . 90 . . . 30

• Answer choices (not foolproof, but a hint)
Four options' lengths are square roots
\(\sqrt{2}\) suggests a 45-45-90 triangle
\(\sqrt{3}\) suggests a 30-60-90 triangle

1) Fill in angle measures
A = C = 120 (opposite angles equal)
A + C = 240
360° = sum of interior angles
360° - 240° = 120° remains, to be split equally:
B = D = 60

2) Draw diagonals AC and BD
=> 4 identical right triangles
Angles: 90° (diagonals' intersection)
60° (bisected A and C)
30° (bisected B and D)

3) Lengths for one triangle
30-60-90 triangles:
sides lengths opposite those angles correspond, respectively,
in ratio \(x: x\sqrt{3}: 2x\)

Diagonal AC\(= (x + x) =\) \(2x\)
Diagonal BD\(= x\sqrt{3} + x\sqrt{3}=\) \(2x\sqrt{3}\)

Ratio of AC: BD?

\(\frac{AC}{BD} = \frac{2x}{2x\sqrt{3}} = \frac{1}{\sqrt{3}}\)

Answer D
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The figure shown is a rhombus in which the measure of angle A = 120°.  [#permalink]

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New post 26 Mar 2018, 06:35
Whenever a Rhombus is formed with internal angles as 120-120-60-60, four 30-60-90 internal triangles can be drawn as diagonals are perpendicular bisectors.

Now, in each 30-60-90 triangle, ratio of height:base is 1:√3.

Hence, Diagonals are in ratio: 2/2*√3 = 1:√3

Hence, Ans D

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Re: The figure shown is a rhombus in which the measure of angle A = 120°.  [#permalink]

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New post 19 May 2018, 10:21
Bunuel wrote:
Image

The figure shown is a rhombus in which the measure of angle A = 120°. What is the ratio of the length of AC to the length of DB?


A. \(1:2\sqrt{3}\)

B. \(1:2\sqrt{2}\)

C. \(1:2\)

D. \(1:\sqrt{3}\)

E. \(1:\sqrt{2}\)


Attachment:
2018-03-21_1019.png
Since diagonals bisect each other at 90 in rhombus and are also angle bisectors. If E is midpoint where AC meets BD.
We get a triangle with 60-30-90.
So AE/DE = 1/sqrt(3)

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Re: The figure shown is a rhombus in which the measure of angle A = 120°. &nbs [#permalink] 19 May 2018, 10:21
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The figure shown is a rhombus in which the measure of angle A = 120°.

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