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In parallelogram ABCD, AB = BD = 2. What is the perimeter of ABCD?

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In parallelogram ABCD, AB = BD = 2. What is the perimeter of ABCD? [#permalink]

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In parallelogram ABCD, AB = BD = 2. What is the perimeter of ABCD?

(A) 2 + 2√2
(B) 2 + 4√2
(C) 4 + 2√2
(D) 4 + 4√2
(E) 12

[Reveal] Spoiler:
Attachment:
2017-11-12_2101_001.png
2017-11-12_2101_001.png [ 5.3 KiB | Viewed 316 times ]
[Reveal] Spoiler: OA

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Kudos [?]: 135380 [1], given: 12691

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Kudos [?]: 397 [0], given: 640

In parallelogram ABCD, AB = BD = 2. What is the perimeter of ABCD? [#permalink]

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New post 12 Nov 2017, 10:19
Bunuel wrote:
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In parallelogram ABCD, AB = BD = 2. What is the perimeter of ABCD?

(A) 2 + 2√2
(B) 2 + 4√2
(C) 4 + 2√2
(D) 4 + 4√2
(E) 12

[Reveal] Spoiler:
Attachment:
The attachment 2017-11-12_2101_001.png is no longer available

Attachment:
cccc.png
cccc.png [ 34.38 KiB | Viewed 196 times ]

Given:
AB = BD = 2
\(\angle\) A = 45°

Parallelogram properties: Opposite sides and opposite angles are equal
Triangle properties: angles opposite equal sides are equal, internal angles sum to 180°

1) Find side AD from properties of isosceles right triangle

\(\triangle\) ABD is an isosceles right triangle
BD = AB, opposite angles are equal
\(\angle\) A = 45° = \(\angle\) ADB
Third \(\angle\) ABD of the triangle = (180 - 45 - 45) = 90°

Length of side AD?

Side length ratio of 45-45-90 triangle*: \(x : x: x\sqrt{2}\)
The two equal sides, which correspond with \(x\), have length 2
Side AD =\(2\sqrt{2}\)

2) Side BC is opposite side AD
AD = BC = \(2\sqrt{2}\)

3) Side CD
CD is opposite AB
CD = AB = \(2\)

4) Perimeter = (\(2 + 2\sqrt{2} + 2 + 2\sqrt{2}\)) =

\(4 + 4\sqrt{2}\)


Answer D

*OR, by Pythagorean Theorem
\(2^2 + 2^2 = (AD)^2\)
\(8 = (AD)^2\)
\(AD = \sqrt{8}=\sqrt{4 * 2}= 2\sqrt{2}\)


**Other properties used to fill in the diagram, though you do not need them if you solve for a 45-45-90 triangle:
Opposite angles A and C are equal and have measures 45°
Opposite angles B and D are equal and have total measures 135°
--Sum of internal angles of parallelogram = 360°
--360 - 45 - 45 = 270°, which must be divided equally between angles B and D = 135° each

Kudos [?]: 397 [0], given: 640

In parallelogram ABCD, AB = BD = 2. What is the perimeter of ABCD?   [#permalink] 12 Nov 2017, 10:19
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