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ABCD is a square with side length 12 ∠CEF is a right angle D

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ABCD is a square with side length 12 ∠CEF is a right angle D [#permalink] New post   04 Jan 2021, 10:47
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\(ABCD\) is a square with side length 12
\(∠CEF\) is a right angle
\(DE = 5\)
\(FB = 9
\)


Quantity A
Quantity B
Perimeter of \(\triangle DEC\)
Perimeter of \(\triangle FEC\)



A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.

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Re: ABCD is a square with side length 12 ∠CEF is a right angle D [#permalink] New post   05 Jan 2021, 13:49
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A) perimeter of triangle DEC is consisted of , DC + DE + EC
B) perimeter of triangle FEC is consisted of , FC + EF + EC

both items includes length of EC, so we can cancel the length of EC to compare

A) DC+DE = 12 + 5 = 17

As FB : BC = 9:12 = 3:4, FC should be 5x3=15
B) FC + EF = 15 + sqrt ( (12-5) ^2 + 3^2 ) = 15 + sqrt(58)

A) 17,
B) 15 + sqrt(58)

sqrt(58) is obviously bigger than 2 which is calculated by 17-5, so B) is bigger.

Answer: B)
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Re: ABCD is a square with side length 12 ∠CEF is a right angle D [#permalink] New post   09 Jan 2021, 23:40
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The question has already been solved above. Mine is more of an observation based solution that requires no calculations.
The triangles in question share EC as a side. The difference being that for DEC, it is the hypotenuse and for FEC, it is not. Now, the hypotenuse is always the biggest side in a triangle and in FEC, CF will be bigger than the other two side. Therefore, the perimeter of FEC will be greater than that of DEC
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Re: ABCD is a square with side length 12 ∠CEF is a right angle D [#permalink] New post   10 Jan 2021, 23:30
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This question involves pythagoreas theorem. After calculating all the sides, we can calculate the perimeters.
dc = 12
de = 5
ec = \sqrt{167}
ef = \sqrt{58}
fc = \sqrt{225}
af = 3
fb = 9
bc = 12
ae = 7

17+ \sqrt{167} [<] 15 + \sqrt{58} + \sqrt{167}
Qb>Qa
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Re: ABCD is a square with side length 12 ∠CEF is a right angle D [#permalink] New post   03 Jun 2021, 03:45
Mrtitsupatsundown wrote:
The question has already been solved above. Mine is more of an observation based solution that requires no calculations.
The triangles in question share EC as a side. The difference being that for DEC, it is the hypotenuse and for FEC, it is not. Now, the hypotenuse is always the biggest side in a triangle and in FEC, CF will be bigger than the other two side. Therefore, the perimeter of FEC will be greater than that of DEC


This is the best approach :blushing:
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Re: ABCD is a square with side length 12 ∠CEF is a right angle D [#permalink]
03 Jun 2021, 03:45
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