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In a rhombus ABCD, AC= 3 units and angle ABC= 120º. What is the area o
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19 Jun 2023, 05:34
19 Jun 2023, 05:34
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In a rhombus ABCD, AC= 3 units and angle ABC= 120º. What is the area of ABCD?(in unit^2)
A. \(\frac{3 \sqrt{3}}{8}\)
B. \(\frac{3 \sqrt{3}}{4}\)
C. \(\frac{3 \sqrt{3} }{2}\)
D. \(3 \sqrt{3}\)
E. \(6 \sqrt{3}\)
Post A Detailed Correct Solution For The Above Questions And Get Kudos.
Question From Our Project: Free GRE Prep Club Tests in Exchange for 20 Kudos
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A. \(\frac{3 \sqrt{3}}{8}\)
B. \(\frac{3 \sqrt{3}}{4}\)
C. \(\frac{3 \sqrt{3} }{2}\)
D. \(3 \sqrt{3}\)
E. \(6 \sqrt{3}\)
Post A Detailed Correct Solution For The Above Questions And Get Kudos.
Question From Our Project: Free GRE Prep Club Tests in Exchange for 20 Kudos
\(\Longrightarrow\) GRE - Quant Daily Topic-wise Challenge
\(\Longrightarrow\) GRE GEOMETRY
Re: In a rhombus ABCD, AC= 3 units and angle ABC= 120º. What is the area o
[#permalink]
16 Jul 2023, 04:37
16 Jul 2023, 04:37
Expert Reply
OE
We know diagonals of a rhombus bisect each other at right angle.
It is given that AC = 3.
Hence, we can say AO = OC = 1.5
(As AC = 3 units)
Also, BO = OD
It is given that ∠ ABC = 1200
Since the diagonals are angle
bisectors in a rhombus,
Hence ∠ABO = ∠OBC = 600
In ∆ ABO,
∠ ABO = 600
∠ AOB = 900
(As diagonals bisect each other at right angle)
Hence, ∠ BAO = 300
We know, in a 300 – 600 – 900
triangle, ratio of the sides is 1:√3: 2
So, in ∆ ABO, ratio of the sides that is, BO: AO: AB is 1:√3: 2
BO: AO: AB
1:3:2
\(\dfrac{1}{\sqrt{3}} : 1 : \dfrac{2}{\sqrt{3}}\)
\(\dfrac{1.5}{\sqrt{3}} : 1.5 : \dfrac{3}{\sqrt{3}}\)
Here, if AO = 1.5, we get BO \(= \dfrac{1.5}{\sqrt{3}}\)
As BO = OD, so OD is \(= \dfrac{1.5}{\sqrt{3}}\)
Therefore \(BD=\dfrac{3}{\sqrt{3}}\)
Hence, we get both diagonals of the rhombus ABCD that is, AC = 3 units and \(BD = \dfrac{3}{\sqrt{3}}\) units.
Area of rhombus = 1/2 × 𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙s
\(\frac{1}{2} \times 3 \times \dfrac{3}{\sqrt{3}}
\)
\(\dfrac{3\sqrt{3}}{2}\)
C is the answer
We know diagonals of a rhombus bisect each other at right angle.
It is given that AC = 3.
Hence, we can say AO = OC = 1.5
(As AC = 3 units)
Also, BO = OD
It is given that ∠ ABC = 1200
Since the diagonals are angle
bisectors in a rhombus,
Hence ∠ABO = ∠OBC = 600
In ∆ ABO,
∠ ABO = 600
∠ AOB = 900
(As diagonals bisect each other at right angle)
Hence, ∠ BAO = 300
We know, in a 300 – 600 – 900
triangle, ratio of the sides is 1:√3: 2
So, in ∆ ABO, ratio of the sides that is, BO: AO: AB is 1:√3: 2
BO: AO: AB
1:3:2
\(\dfrac{1}{\sqrt{3}} : 1 : \dfrac{2}{\sqrt{3}}\)
\(\dfrac{1.5}{\sqrt{3}} : 1.5 : \dfrac{3}{\sqrt{3}}\)
Here, if AO = 1.5, we get BO \(= \dfrac{1.5}{\sqrt{3}}\)
As BO = OD, so OD is \(= \dfrac{1.5}{\sqrt{3}}\)
Therefore \(BD=\dfrac{3}{\sqrt{3}}\)
Hence, we get both diagonals of the rhombus ABCD that is, AC = 3 units and \(BD = \dfrac{3}{\sqrt{3}}\) units.
Area of rhombus = 1/2 × 𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙s
\(\frac{1}{2} \times 3 \times \dfrac{3}{\sqrt{3}}
\)
\(\dfrac{3\sqrt{3}}{2}\)
C is the answer
gmatclubot
Re: In a rhombus ABCD, AC= 3 units and angle ABC= 120º. What is the area o [#permalink]
16 Jul 2023, 04:37
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