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

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Joined: 15 Nov 2018
Posts: 22
Location: United States
Concentration: Finance, Entrepreneurship
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Triangles AED and BEC are formed using the straight lines AB and CD as  [#permalink]

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Updated on: 15 Jan 2019, 11:12
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Difficulty:

95% (hard)

Question Stats:

37% (03:48) correct 63% (03:15) wrong based on 34 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.
Intern
Joined: 11 Dec 2018
Posts: 9
Re: Triangles AED and BEC are formed using the straight lines AB and CD as  [#permalink]

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14 Jan 2019, 14:19

Posted from my mobile device
Intern
Joined: 15 Nov 2018
Posts: 22
Location: United States
Concentration: Finance, Entrepreneurship
GPA: 3.76
Triangles AED and BEC are formed using the straight lines AB and CD as  [#permalink]

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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.

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

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.

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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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