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Math Expert V
Joined: 02 Sep 2009
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17 00:00

Difficulty:   85% (hard)

Question Stats: 58% (03:12) correct 42% (02:33) wrong based on 64 sessions

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If on the coordinate plane $$(6, 2)$$ and $$(0, 6)$$ are the endpoints of the diagonal of a square, what is the distance between point $$(0, 0)$$ and the closest vertex of the square?

A. $$\frac{1}{\sqrt{2}}$$
B. 1
C. $$\sqrt{2}$$
D. $$\sqrt{3}$$
E. $$2\sqrt{3}$$

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Math Expert V
Joined: 02 Sep 2009
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3
Official Solution:

If on the coordinate plane $$(6, 2)$$ and $$(0, 6)$$ are the endpoints of the diagonal of a square, what is the distance between point $$(0, 0)$$ and the closest vertex of the square?

A. $$\frac{1}{\sqrt{2}}$$
B. 1
C. $$\sqrt{2}$$
D. $$\sqrt{3}$$
E. $$2\sqrt{3}$$

Given endpoints of diagonal of a square: $$B(0,6)$$ and $$D(6,2)$$. Let other vertices be $$A$$ (closest to the origin) and $$C$$ (farthest to the origin): Length of the diagonal would be: $$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}=\sqrt{(6-0)^2+(2-6)^2}=\sqrt{52}$$

Coordinates of the midpoint $$M$$ of the diagonal would be: $$M(x,y)=(\frac{6+0}{2},\frac{2+6}{2})=(3,4)$$.

{The slope of line segment AM}*{The slope of line segment BD}=-1 (as they are perpendicular to each other), so $$\frac{y-4}{x-3}*\frac{6-2}{0-6}=-1$$, which simplifies to $$y-4=\frac{3}{2}(x-3)$$

The distance between the unknown vertices to the midpoint is half the diagonal:

$$(x-3)^2+(y-4)^2=(\frac{\sqrt{52}}{2})^2=13$$;

$$(x-3)^2+\frac{9}{4}(x-3)^2=13$$;

$$(x-3)^2=4$$;

Which gives: $$x=1$$ and $$y=1$$ OR $$x=5$$ and $$y=7$$.

Hence point $$A$$ is at $$(1,1)$$ and point $$C$$ is at $$(5,7)$$. Closest to the origin is $$A$$. Distance $$OA=\sqrt{2}$$

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Hi akhil911, maybe this will help. I calculated the midpoint in the same way as described by Bunuel to find (3,4). Then I determined the delta from point B to the midpoint M as (0,6) - (3,4) = (3,2). In order to find point A simply swap the x and y and subtract from the midpoint: (3,4) - (2,3) = (1,1). Which gives √2 as distance.
##### General Discussion
Manager  Joined: 11 Aug 2011
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Hi

Is there any shorter way to solve this problem.
I have understood the solution but this seems to be a very tough and complex problem.
I am wondering if this can be solved in some other way.
Intern  B
Joined: 04 Aug 2014
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1
Or for points (6,2) and (0,6) in this case:
Absolute difference for x is 6-0=6
Absolute difference for y is 6-2=4
Using logic, since this is a square, sides need to be equal and average of 6+4 is 5
Hence, absolute difference of 5 units away is for x is x=1 and for y is y=1, with intersection at point (1, 1) being equidistant to B and D
Then work out the distance as before
Manager  Joined: 18 May 2016
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Shorter solution if one can draw a correct sketch of the figure:

The shortest distance from (0,0) to one of the vertexes of the square is perpendicural distance to A. A and (0,0) forms a square with s=1. The shortest distance will be the diagonal of that square which is V2s or 1 x V2 = V2 the answer.

Even if the graph is drawn somewhat right, we are looking for the answer that has nV2 in it.
Manager  B
Joined: 25 Sep 2015
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The distance between the unknown vertices to the midpoint is half the diagonal:

(x−3)2+(y−4)2=(52√2)2=13(x−3)2+(y−4)2=(522)2=13;

I don't understand what this step is doing?
Current Student B
Joined: 19 Mar 2013
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rachitshah
It seems like the steps from the given explanation accomplish the following:
1) Find the distance of the original diagonal: (6-0)^2 + (2-6)^2 = X^2; X = root 52
2) Find the midpoint of the original diagonal: (6+0)/2, (6+2)/2 = (3,4)
3) Find the equation for the line for the other diagonal, since this will be the line that has the coordinates closest to (0,0)
Key to this step: This line is the perpendicular bisector of the diagonal (the diagonals of squares are perpendicular bisectors = fact in geometry)
You can find the equation for the line the other diagonal by doing the following:
a) y = mx + b (generic equation for line)
b) find the slope of the original diagonal: (6-2)/(0-6) = -2/3 and take the negative reciprocal of this = 3/2 since the other diagonal is perpendicular
c) now you have y = 3x/2 + b; find the value of b by plugging in (3,4) and you get y = 3x/2 - 1/2, which is the same thing as y-4 = 3/2(x-3)
4) Distance from farthest coordinates to midpoint of diagonal = Distance from shortest coordinates to midpoint of diagonal = 1/2 the length of the diagonal (in this case, (root52/2)^2).
5) The distance from the farthest or shortest coordinates of the diagonal to the midpoint can be solved by using pythagorean theorem:
(x-3)^2 +(y-4)^2 = (root52/2)^2. (root52/2)^2 = 13
6) From step 3 you know that y-4 = 3/2(x-3). Plug this into y for step 5 above ==> (x-3)^2 + (y-4)^2 = 13 --> (x-3)^2 + [3/2(x-3)]^2 = 13 -->
(x-3)^2 +9/4(x-3)^2 = 13; solving for x gives you (x-3)^2 = 4, so x can be 1 or 5.
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I think this is a high-quality question and I agree with explanation.
Intern  Joined: 11 Nov 2014
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rachitshah wrote:
The distance between the unknown vertices to the midpoint is half the diagonal:

(x−3)2+(y−4)2=(52√2)2=13(x−3)2+(y−4)2=(522)2=13;

I don't understand what this step is doing?

This step is finding the distance from vertex A(x,y) to mid point M(3,4) or the length of AM you can say.
since AC is also a diagonal, AM will be half its length (which is\sqrt{52} and diagonals of a square are equal in length.)

I hope it clears your doubt.
Manager  G
Joined: 27 Jan 2016
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Skytail wrote:
Hi akhil911, maybe this will help. I calculated the midpoint in the same way as described by Bunuel to find (3,4). Then I determined the delta from point B to the midpoint M as (0,6) - (3,4) = (3,2). In order to find point A simply swap the x and y and subtract from the midpoint: (3,4) - (2,3) = (1,1). Which gives √2 as distance.

Can you pls explain the logic behind this approach. What do you mean by delta here.
Manager  G
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Bunuel wrote:
Official Solution:

If on the coordinate plane $$(6, 2)$$ and $$(0, 6)$$ are the endpoints of the diagonal of a square, what is the distance between point $$(0, 0)$$ and the closest vertex of the square?

A. $$\frac{1}{\sqrt{2}}$$
B. 1
C. $$\sqrt{2}$$
D. $$\sqrt{3}$$
E. $$2\sqrt{3}$$

Given endpoints of diagonal of a square: $$B(0,6)$$ and $$D(6,2)$$. Let other vertices be $$A$$ (closest to the origin) and $$C$$ (farthest to the origin): Length of the diagonal would be: $$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}=\sqrt{(6-0)^2+(2-6)^2}=\sqrt{52}$$

Coordinates of the midpoint $$M$$ of the diagonal would be: $$M(x,y)=(\frac{6+0}{2},\frac{2+6}{2})=(3,4)$$.

{The slope of line segment AM}*{The slope of line segment BD}=-1 (as they are perpendicular to each other), so $$\frac{y-4}{x-3}*\frac{6-2}{0-6}=-1$$, which simplifies to $$y-4=\frac{3}{2}(x-3)$$

The distance between the unknown vertices to the midpoint is half the diagonal:

$$(x-3)^2+(y-4)^2=(\frac{\sqrt{52}}{2})^2=13$$;

$$(x-3)^2+\frac{9}{4}(x-3)^2=13$$;

$$(x-3)^2=4$$;

Which gives: $$x=1$$ and $$y=1$$ OR $$x=5$$ and $$y=7$$.

Hence point $$A$$ is at $$(1,1)$$ and point $$C$$ is at $$(5,7)$$. Closest to the origin is $$A$$. Distance $$OA=\sqrt{2}$$

Hi Bunuel,

Is there any shorter way to solve this problem. Also, can you pls explain the logic behind the Skytail's approach in one of the above posts.
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this is a very very lengthy problem!
It's great for practice though!
Intern  B
Joined: 18 Dec 2015
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Quicker way to solve this:

The midpoint of the diagonal is (3,4). The vector from the right bottom corner to the midpoint is V1 = <-3, -2>. You need to find a vector perpendicular to this vector and of the same magnitude and add it to the midpoint position vector to find the corners.

Vectors perpendicular meet:

-3x - 2y = 0 or y = (3/2) x
So the form is <x, 3/2x>

The two perpendicular vectors that have the same magnitude as V1 will meet:

x^2 + [(3/2)x]^2 = 13
x = +/- 2

So the two vectors are <2,3> or <-2,-3>

To find the corner closest to the origin add <-2,-3> to the position vector of the midpoint <3,4> to get <1,1>.

The magnitude of this vector sqrt(2) is the answer.
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akhil911 wrote:
Hi

Is there any shorter way to solve this problem.
I have understood the solution but this seems to be a very tough and complex problem.
I am wondering if this can be solved in some other way.

$$m = \frac{(y2 - y1)}{(x2-x1)}$$

$$m = \frac{6-2}{0-6} = \frac{-2}{3}$$

The other diagonal of a square is perpendicular to this one and we know slope of the perpendicular line is always $$\frac{-1}{m} = \frac{-1}{-2/3} = \frac{3}{2}$$

Now mid point of two coordinates or the diagonal of the square is $$(3,4)$$.

$$Slope = \frac{rise}{run}$$

From mid point of $$(3,4)$$, we need to back traverse a little. Since line is rising 3 and running 2, we reduce rise by 3 and run by 2 while back tracking to come up with the point $$(3-1,4-3)$$ i.e. $$(1,1)$$ which is the vertex of the square nearest to $$(0,0)$$

$$(0,0)$$ and $$(1,1)$$ be solved using Pythagoras theorem. $$\sqrt{1^2 + 1^2}$$ and this comes out to be $$\sqrt{2}$$
Intern  B
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I approached this problem more visually. I drew the two points, forming a diagonal. Then knowing that this is a square, I know that the intersection of the two diagonals must be perpendicular. I find the slope of the diagonal given (2/3) so I flip and make this negative to find the slope of the diagonal (-3/2). I used the midpoint formula to find the center of the square, or the diagonal intersection. That value is (3,4). So from 3,4 I move down 3 and over 2. That is the end of my other diagonal - I can't go further or else it won't be a square. I now have my figure and four endpoints.

Without doing any math, I know that the distance from 0,0 to 1,1 is root(2). I think leveraging the grid and drawing out the exact coordinates really helps with this problem.
Veritas Prep GMAT Instructor V
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Bunuel wrote:
If on the coordinate plane $$(6, 2)$$ and $$(0, 6)$$ are the endpoints of the diagonal of a square, what is the distance between point $$(0, 0)$$ and the closest vertex of the square?

A. $$\frac{1}{\sqrt{2}}$$
B. 1
C. $$\sqrt{2}$$
D. $$\sqrt{3}$$
E. $$2\sqrt{3}$$

Responding to a pm:

Here:

http://www.veritasprep.com/blog/2013/04 ... ry-part-i/
This post will explain you the relevant concept.

and this post: https://www.veritasprep.com/blog/2013/0 ... y-part-ii/
discusses this question in detail.
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Few additional inputs to those which are mentioned above and maybe a graphical presentation will help understand the solution in a better way.

Additional points:- As M is the mid point of BD, in the same way M is the mid point of XY.

Solution ----> we come from M------>Y------->A (with coordinates 1,1)

Attachment:
Coord Geo Soln.jpg

Attachment:
Coord Geo Soln 2.jpg

Attachment:
Coord Geo Soln 3.jpg

>> !!!

You do not have the required permissions to view the files attached to this post.

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Here's my 2 cents' worth:

1. Given the coordinates, you can find immediately 2 things:
a) Midpoint = (3,4) - just by doing some quick mental math;
b) Gradient = (-2/3) - again, do a quick calculation using the 2 points given.

2. We know that the unknown vertex MUST fall along the other diagonal passing through the midpoint (3,4) i.e. the diagonal whose gradient is perpendicular to what we just calculated.
------> Gradient of other diagonal = 3/2

3. The unknown vertices, when calculating gradient in relation to the midpoint, MUST yield 3/2.
- We also know that the square is in Quadrant I (top right).
- Observe that:

a) (y-4) / (x-3) = 3/2 --> pick 2 numbers:
- (7-4 = numerator; 5-3 = denominator) --> (5,7) works.
- Alternatively, we also know (1-4)/(1-3) gives us -3/-2 = 3/2 (bingo).

b) (1,1) is what we need since it's the closest to the answer.

Distance is simply the hypotenuse of a right-angled isoceles triangle on the graph paper (1 square grid of 1x1) = Root 2. --> Answer = C M15-19   [#permalink] 28 Nov 2019, 06:51
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# M15-19

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