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

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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
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