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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
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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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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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Joined: 11 Dec 2018
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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: 21
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.

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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