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# If ABCD is a rectangle, then what is the length of EC?

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Manager
Joined: 03 Mar 2018
Posts: 182
If ABCD is a rectangle, then what is the length of EC?  [#permalink]

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15 Mar 2018, 12:12
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Difficulty:

35% (medium)

Question Stats:

69% (02:43) correct 31% (03:34) wrong based on 27 sessions

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If ABCD is a rectangle, then what is the length of EC?
Attachment:

Capture1.JPG [ 16.74 KiB | Viewed 377 times ]

(A) 7.8
(B) 8
(C) 8.4
(D) 9
(E) 9.6

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Intern
Joined: 06 Oct 2017
Posts: 9
If ABCD is a rectangle, then what is the length of EC?  [#permalink]

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15 Mar 2018, 15:37
From Pythagorean theorem the diagonal $$BD = \sqrt{12^2 + 16^2} = \sqrt{400} = 20$$
Let $$ED = x$$, and $$EB = 20 - x$$

Then, we have two equations:
(1) $$EC^2 + (20-x)^2 = 16^2$$
(2) $$EC^2 + x^2 = 12^2$$

Solving equation 2 we get:
$$EC^2 + 400 - 40x + x^2 = 256$$
$$EC^2 +x^2 - 40x = -144$$

Substracting equation (1) from (2) results in:
$$40x = 288; x = 7.2$$

Then, from triangle DEC -
$$7.2^2 + EC^2 = 12^2$$
$$51.84 + EC^2 = 144$$
$$EC = \sqrt{92.16}$$

We do not need to calculate we can use the answers. $$\sqrt{92.16}$$ is obviously more than 9, since $$9^2 = 81$$ hence the only answer suitable is $$EC = 9.6$$

Intern
Joined: 17 Jan 2018
Posts: 44
Re: If ABCD is a rectangle, then what is the length of EC?  [#permalink]

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15 Mar 2018, 18:48
We can see that Tr BAD is a right angled triangle with two sides 12 and 16.
Now, we know (3, 4, 5) as a set of sides for a right angled triangle. Multiply each by 4, we get (12, 16, 20)

So, the side BD is 20.

Area of Tr BCD is half of the area of Rec ABCD

1/2*20*EC = 1/2*12*16

EC = 9.6cm

itisSheldon wrote:
If ABCD is a rectangle, then what is the length of EC?
Attachment:
Capture1.JPG

(A) 7.8
(B) 8
(C) 8.4
(D) 9
(E) 9.6
Re: If ABCD is a rectangle, then what is the length of EC? &nbs [#permalink] 15 Mar 2018, 18:48
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