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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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12 Nov 2017, 10:11
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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 [ 5.3 KiB | Viewed 872 times ]
[Reveal] Spoiler: OA

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Joined: 22 May 2016
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In parallelogram ABCD, AB = BD = 2. What is the perimeter of ABCD? [#permalink]

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12 Nov 2017, 11:19
Bunuel wrote:

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 [ 34.38 KiB | Viewed 626 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°

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 = ($$2 + 2\sqrt{2} + 2 + 2\sqrt{2}$$) =

$$4 + 4\sqrt{2}$$

*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

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In parallelogram ABCD, AB = BD = 2. What is the perimeter of ABCD?   [#permalink] 12 Nov 2017, 11:19
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