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# In the figure above, ABCD is a square. What are the coordinates of poi

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Math Expert
Joined: 02 Sep 2009
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In the figure above, ABCD is a square. What are the coordinates of poi  [#permalink]

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03 Jul 2018, 09:15
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In the figure above, ABCD is a square. What are the coordinates of point B?

(A) (-4,2)
(B) (-2,4)
(C) (-2,6)
(D) (4, -6)
(E) (6,-2)

Attachment:

square.jpg [ 12.91 KiB | Viewed 4285 times ]

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Re: In the figure above, ABCD is a square. What are the coordinates of poi  [#permalink]

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03 Jul 2018, 10:48
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Another approach:

Diagonals will intersect at midpoints of the two lines. Let B be (x,y)

So, (x + 0)/2 = (4 + (-6))/2 which gives x = -2
(y + (-4))/2 = (2+0)/2 which gives y = 6

Hence B is (-2,6), Option C.
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Re: In the figure above, ABCD is a square. What are the coordinates of poi  [#permalink]

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03 Jul 2018, 09:44
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Bunuel wrote:

In the figure above, ABCE is a square. What are the coordinates of point B?

(A) (-4,2)
(B) (-2,4)
(C) (-2,6)
(D) (4, -6)
(E) (6,-2)

Attachment:
square.jpg

ABCD is a square as per the given geometrical figure. ("ABCE" may be a typo error)

AB || CD
Or, slope of line AB=Slope of line CD
Or, $$\frac{y-0}{x-(-6)}$$=$$\frac{2-(-4)}{4-0}$$ (Co-ordinates of vertex B is (x,y) say)
Or,$$y=\frac{3x+18}{2}$$------------(1)
Since vertex B lies in $$Q_{2}$$, so x<0 & y>0
Option D & E are eliminated.
Now, among options A, B, and C; option C satisfies the eq(1) above.

Therefore. Ans(C).
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Re: In the figure above, ABCD is a square. What are the coordinates of poi  [#permalink]

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24 Jul 2018, 11:20
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Another Approach:

Since the distance between each of the points, A, B , C and D should be same..So we know that the distance between each of the points should be \sqrt{6^2+4^2}. And only one option (-2,6) gives us that distance.
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Re: In the figure above, ABCD is a square. What are the coordinates of poi  [#permalink]

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25 Jul 2018, 17:45
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Bunuel wrote:

In the figure above, ABCE is a square. What are the coordinates of point B?

(A) (-4,2)
(B) (-2,4)
(C) (-2,6)
(D) (4, -6)
(E) (6,-2)

We can get the answer without calculating anything except two obvious lengths,
using either elimination or the "box" method

Eliminate:
• Answers D) (4, -6) and E) (6,-2). Coordinates of B are in Q II (-x,y). D and E have (x, -y)

• Answer A: (-4, 2) is the mirror point for C (4, 2). Not allowed. Square is not symmetric about the axes.
If you were to flip C across the y-axis, its mirror point would be (-4, 2)
C's mirror point across the y-axis cannot be a vertex because the square is not symmetric about the axes.

• Answer B: (-2,4). Distance of B from x-axis MUST be greater than distance of D from x-axis. More than half of the square is in QI and QII.
The y-coordinate of B (the height of B from the x-axis) MUST be greater than D's distance from x-axis. D's distance is 4. B must be > 4

That leaves one answer: (-2, 6)

Attachment:

square2018.07.25.jpg [ 35.18 KiB | Viewed 3332 times ]

The "box method"

If geometric figures are not parallel to the x- and y-axes, draw a box around the figure.

Now we have congruent right triangles all around the smaller square.

The side lengths of those triangles are easy to find because they are right triangles whose vertical and horizontal legs can be measured with the x- and y-axes.

From line segment AO = 6 and properties of a square (parallel sides) we know that one segment of a side of large blue-edged square = 6

From line segment DO = 4 and properties of a square we know that the other segment of a blue-edged side = 4

A square inscribed in a square divides the sides of the larger square proportionally
(Pythagorean theorem)

The larger square's sides are in segments of length 6, 4

Right triangle ABX has height 6. By properties of a right triangle (perpendicular legs), point X and point B are collinear.

The y-coordinate of B is 6. That leaves one answer.
(-2, 6)

If you wanted to ascertain the x-coordinate, keep in mind the larger square side's a, b, a, b pattern

By properties of a square, Points C and Y MUST have the same x-coordinate.
Both are 4 away from the y-axis.
But segment BY must have length 6 (by property of inscribed square)
In QII, then, the distance of B from the y-axis is (6-4) = 2

Because the vertex is in QII, its x-coordinate is (-2, 6)
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Re: In the figure above, ABCD is a square. What are the coordinates of poi  [#permalink]

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30 Jul 2018, 08:20
Bunuel wrote:

In the figure above, ABCE is a square. What are the coordinates of point B?

(A) (-4,2)
(B) (-2,4)
(C) (-2,6)
(D) (4, -6)
(E) (6,-2)

Attachment:
square.jpg

We have a right triangle in the third quadrant which sides are 4 and 6, the hypotenuse is 4^2 + 6^2 = \sqrt{52}
So, if we would draw a triangle in the forth quadrant the hypotenuse would be \sqrt{52}, the side parallel to the Y axes sould be 6, just option B has this value.

Option B.
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Re: In the figure above, ABCD is a square. What are the coordinates of poi  [#permalink]

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03 Aug 2018, 08:45
1
AB and DC are parallel, which means the slopes are identical. Also, distances AB = DC.

This means that the change in the x-coordinate and y-coordinate from D to C will be the same as the change from A to B.

x-coordinate of D increases by 4 to reach x-coordinate of C
y-coordinate of D increases by 6 to reach y-coordinate of C

so:

x-coordinate of A must increase by 4 to reach x-coordinate of B: -6+4=-2
y-coordinate of A must increase by 6 to reach y-coordinate of B: 0+6=6

Therefore the coordinates of B are (-2,6). Answer B.
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Re: In the figure above, ABCD is a square. What are the coordinates of poi  [#permalink]

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11 Aug 2018, 08:00
1
Bunuel wrote:

In the figure above, ABCE is a square. What are the coordinates of point B?

(A) (-4,2)
(B) (-2,4)
(C) (-2,6)
(D) (4, -6)
(E) (6,-2)

Attachment:
square.jpg

Dear Moderator ,
There seems to be a small typo here, Square ABCE has come up instead of square ABCD. Hope you will do the needful, Thank you.
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Posts: 58464
Re: In the figure above, ABCD is a square. What are the coordinates of poi  [#permalink]

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11 Aug 2018, 14:01
stne wrote:
Bunuel wrote:

In the figure above, ABCE is a square. What are the coordinates of point B?

(A) (-4,2)
(B) (-2,4)
(C) (-2,6)
(D) (4, -6)
(E) (6,-2)

Attachment:
square.jpg

Dear Moderator ,
There seems to be a small typo here, Square ABCE has come up instead of square ABCD. Hope you will do the needful, Thank you.

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Edited. Thank you.
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Re: In the figure above, ABCD is a square. What are the coordinates of poi  [#permalink]

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12 Aug 2018, 06:24
Bunuel wrote:

In the figure above, ABCD is a square. What are the coordinates of point B?

(A) (-4,2)
(B) (-2,4)
(C) (-2,6)
(D) (4, -6)
(E) (6,-2)

Attachment:
square.jpg

Another one !!
since it is a square , the distance between any two points should be same .
distance between c and d = \sqrt{42}
so
\sqrt{42}= \sqrt{(4-x)^2+(2-y)^2}
C satisfies the equation for x and y
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Re: In the figure above, ABCD is a square. What are the coordinates of poi  [#permalink]

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16 Aug 2018, 11:05
1
You all give very fine mathematical solutions. Let me give you an alternative that took me 20s.
Measure distance from (0,0) to (-6, 0) with my pen. Compare it point B. Seems like x=-2 and Y=6. Check options: Answer is C.
PS questions are always drawn to scale unless stated otherwise.

+1 if you think this was useful.
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Re: In the figure above, ABCD is a square. What are the coordinates of poi  [#permalink]

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22 Aug 2018, 05:26
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ErikLewe wrote:
You all give very fine mathematical solutions. Let me give you an alternative that took me 20s.
Measure distance from (0,0) to (-6, 0) with my pen. Compare it point B. Seems like x=-2 and Y=6. Check options: Answer is C.
PS questions are always drawn to scale unless stated otherwise.

+1 if you think this was useful.

That solution should be absolutely fine. I read this approach in the Princeton review book. So it should be approved .
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Re: In the figure above, ABCD is a square. What are the coordinates of poi  [#permalink]

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29 Aug 2019, 22:31
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Re: In the figure above, ABCD is a square. What are the coordinates of poi   [#permalink] 29 Aug 2019, 22:31
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