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Triangles AED and BEC are formed using the straight lines AB and CD as

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Triangles AED and BEC are formed using the straight lines AB and CD as  [#permalink]

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New post Updated on: 15 Jan 2019, 11:12
1
3
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

34% (03:48) correct 66% (03:18) wrong based on 36 sessions

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Triangles AED and BEC are formed using the straight lines AB and CD as shown in the figure above. If \(BE = BC, DE^2 > AD^2 + AE^2\) and the measure of \(∠AED\) is x, which of the following statements must be true?

I. \(CE^2 > 2BE^2\)
II. \(AE < AD\)
III. \(DE > CE\)


A) I only

B) II only

C) III only

D) I, II and III

E) None of the above

Originally posted by zubair123 on 14 Jan 2019, 11:09.
Last edited by zubair123 on 15 Jan 2019, 11:12, edited 1 time in total.
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Re: Triangles AED and BEC are formed using the straight lines AB and CD as  [#permalink]

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New post 14 Jan 2019, 14:19
Explanation please

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Triangles AED and BEC are formed using the straight lines AB and CD as  [#permalink]

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New post 15 Jan 2019, 11:12
EXPLANATION:

1) Labeling the figure from given information:

Let's label the figure from the given information:

1.1) \(∠DAE\) = (180-3x) from lower triangle

1.2) Since E is the intersection point of two lines, \(∠BEC\) = \(∠AED\) = x

1.3) Since BE = BC, \(∠ECB\) = \(∠BCE\) = x

1.4) \(∠EBC\) = 180-2x

1.5) Since \(DE^2 > AD^2 + AE^2\), we know that angle opposite to side DE is > 90 degrees and it is an obtuse triangle. So angle opposite to DE is \(∠DAE\) which from 1.1 is 180-3x.

So 180-3x >90, and x <30

2) Evaluating each statement independently:

Now that we have established our pre-thinking, let's evaluate each statement:

I) \(CE^2 > 2BE^2\) which is the same as \(CE^2 > BE^2 + BC^2\) since BE = BC

Now for the above inequality to be true, angle opposing CE should be greater than 90, which means 190-2x should be greater than 90 , we already know that x <30, so 180-2x should be less than 120.

Hence sufficient condition. I is true.

II) AE <AD

we know that angle opposing AD is greater than angle opposing side AE, so the above inequality cannot be true.

III) AE > AD

we know that angles opposing each side is proportional to that side length. So angle opposing AE = 180-3x and angle opposing AD is 180-2x. Under all circumstances. 180-3x < 180-2x,

hence Statement III is not true.

So answer is I only. Hence A is the answer.

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Triangles AED and BEC are formed using the straight lines AB and CD as   [#permalink] 15 Jan 2019, 11:12
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