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11 Dec 2017, 23:10
Kudos
Bookmarks
B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
64% (02:11) correct
36%
(01:48)
wrong
based on 158
sessions
History
Date
Time
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Not Attempted Yet
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.
Attachment:
2017-12-12_1002_001.png [ 6.75 KiB | Viewed 5321 times ]
12 Dec 2017, 12:15
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Bunuel
Attachment:
eeee.png [ 18.47 KiB | Viewed 4782 times ]
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:
\(4\sqrt{2} + 4\sqrt{2} + 5 + 5 =\) \((10 + 8√2)\): Answer B
04 Feb 2020, 05:23
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Bookmarks
Bunuel
Solution
- • Perimeter of parallelogram \(OABC = OA+AB+BC+CO \)
- o Since OA = BC and AB = CO hence, perimeter of OABC \(= 2 *(BC+OC)\)
- o We can observe from the diagram, BE = distance of point B from x axis = 4
• Since OA is parallel to CB hence, angle BCE is 45°
- o triangle CEB is a 45° -90°- 45° triangle.
- So, ratio of the sides EC: BC:BE \(= 1: \sqrt{2}: 1\)
- \(BC = \sqrt{2} * BE = 4*\sqrt{2}\)
And, \(EC = BE = 4\)
- o So, \(OC = OE – EC = 9-4 = 5\)
