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In the xy plane, the rectangular ABCD is placed such that AB is paral

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In the xy plane, the rectangular ABCD is placed such that AB is paral  [#permalink]

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New post 27 Jun 2020, 09:26
3
2
00:00
A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

40% (02:17) correct 60% (02:48) wrong based on 35 sessions

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In the xy-plane, the rectangle ABCD is placed such that AB is parallel to the x-axis and CB is the parallel to the y-axis. A right triangle, ABE not shown, is drawn inside the rectangle, such that side AB is the hypotenuse of the triangle, and point E is on side CD of the rectangle. Which of the following could be the x-coordinate of point E?

A) 2
B) 2.5
C) 3
D) 4
E) 5.25
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Re: In the xy plane, the rectangular ABCD is placed such that AB is paral  [#permalink]

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New post 27 Jun 2020, 10:18
Y cordinate of E will be 3
Let X cordinate be x
Length of AB = 5

Length of one leg of right angle triangle = √(x-1)^2+(3-1)^2

Length of other leg of right angle triangle = √(x-6)^2+(3-1)^2

AB^2 = (x-1)^2+(3-1)^2 + (x-6)^2+(3-1)^2
25 = (x-1)^2 + 4 + (x-6)^2 + 4
17 = (x-1)^2 + (x-6)^2
17 = x^2+1-2x+x^2+36-12x
17-37 = 2X^2-14x
-20 = 2X^2-14x
-10 = X^2-7x
0 = X^2-7x+10
0 = X^2-5x-2x+10
0 = X(X-5)-2(X-5)
0 = (X-5)(X-2)
X = 5&2
Since 5 is not among options.
2 is the answer

Answer is option A

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Re: In the xy plane, the rectangular ABCD is placed such that AB is paral  [#permalink]

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New post 27 Jun 2020, 21:42
2
1
IMO -A
Slope of perpendicular line is
M1*M2=-1
Slope of AE =(3-1)/(x-1)
Slope of BE =(1-2)/(6-x)

-1= 2/(x-1) *-2/(6-x)
X^2-7x+10=0
(X-5)(x-2)=0
X=2 &5
Only 2 is available in the option so answer is A

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Re: In the xy plane, the rectangular ABCD is placed such that AB is paral  [#permalink]

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New post 27 Jun 2020, 21:57
Explanation:

We can easily solve this question by Options.

Checking Option A: x cordinate = 2
We know, AB=5 , Elies on Line CD (i.e. Y coordinate is 3).
So E (2,3)
Given: A(1,1) , B(6,1). ABCD is rectangle. C(6,3) D(1,3)
AE = √(2^2 + 1^2) = √5
BE= √(4^2 + 2^2) = √20
As ABE is right angle triangle
So, AE^2 + BE^2 = AB^2
(√5)^2 +(√20)^2 = AB^2
AB=√25 = 5

IMO-A
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Re: In the xy plane, the rectangular ABCD is placed such that AB is paral  [#permalink]

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New post 07 Jul 2020, 02:00
is something wrong with the question? how can CD be parallel to the Y axis ? it should be parallel to x axis or AC should be parallel to Y-axis. Or m I interpreting it incorrectly ?
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Re: In the xy plane, the rectangular ABCD is placed such that AB is paral  [#permalink]

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New post 12 Jul 2020, 22:08
VERBAL1 wrote:
is something wrong with the question? how can CD be parallel to the Y axis ? it should be parallel to x axis or AC should be parallel to Y-axis. Or m I interpreting it incorrectly ?




Hi VERBAL1,

There was a minor typo in the question, It has now been corrected.
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Re: In the xy plane, the rectangular ABCD is placed such that AB is paral   [#permalink] 12 Jul 2020, 22:08

In the xy plane, the rectangular ABCD is placed such that AB is paral

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