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In the figure above, what is the perimeter of parallelogram OABC?

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In the figure above, what is the perimeter of parallelogram OABC? [#permalink]

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11 Dec 2017, 22:10
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In the figure above, what is the perimeter of parallelogram OABC?

(A) 10 + 4√2
(B) 10 + 8√2
(C) 18
(D) 36
(E) It cannot be determined from the information given.

[Reveal] Spoiler:
Attachment:

2017-12-12_1002_001.png [ 6.75 KiB | Viewed 259 times ]
[Reveal] Spoiler: OA

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In the figure above, what is the perimeter of parallelogram OABC? [#permalink]

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12 Dec 2017, 11:15
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Bunuel wrote:

In the figure above, what is the perimeter of parallelogram OABC?

(A) 10 + 4√2
(B) 10 + 8√2
(C) 18
(D) 36
(E) It cannot be determined from the information given.

[Reveal] Spoiler:
Attachment:
The attachment 2017-12-12_1002_001.png is no longer available

Attachment:

eeee.png [ 18.47 KiB | Viewed 103 times ]

Drop an altitude from vertex B to x-axis, point N
Draw a line from vertex A to y-axis, point M

∠ ABC = 45° (it is opposite congruent parallelogram ∠ O = 45°)

Derive:
Height is 4
-- y-coordinate of vertex B = 4
-- y-coordinate of N = 0
-- MB || ON, and BN is ⍊ to both, so distance between B and N = height of parallelogram =
-- difference between y-coordinates: (4 - 0) = 4
Height = 4 = Side BN

Triangle BCN is isosceles right triangle, from which we find two side lengths

∆ BCN has one right angle and two 45° angles
-- right angle at N
-- ∠ C = 45° ( ∠ O = 45° = ∠ C, angles are corresponding in parallel lines cut by a transversal)
-- and other ∠ CBN = 45°
(Right angle at B) - (∠ABC) = ∠CBN
(90 - 45) = 45°

45-45-90: ∆ BCN is right isosceles

Sides opposite equal angles (45°) are equal
BN = CN = 4
Sides are in ratio $$x : x : x\sqrt{2}$$
Length $$4$$ corresponds with $$x$$
Hypotenuse BC corresponds with $$x\sqrt{2}$$

BC = $$4\sqrt{2}$$ = OA

Side lengths: AB and OC

Side OC length = 5 = AB
-- Point N coordinates are (9,0)
Points B and N lie on parallel lines; a perpendicular line from B intercepts opposite side at x-coordinate of B, so N = (9,0)

-- Point C coordinates must be (5,0)
(x-coordinate of N, 9) - (side length, 4) = x-coordinate of C
(9 - 4) = 5 = x-coordinate of C

-- Vertices O and C lie on same line; subtract x-coordinates to get distance from C to O
(5-0) = 5 = length of OC

OC = AB = 5

Perimeter:
[Reveal] Spoiler:
$$4\sqrt{2} + 4\sqrt{2} + 5 + 5 =$$ $$(10 + 8√2)$$: Answer B

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In the figure above, what is the perimeter of parallelogram OABC?   [#permalink] 12 Dec 2017, 11:15
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In the figure above, what is the perimeter of parallelogram OABC?

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