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I've been having a hard time understanding the solution to this problem. Can anyone offer a simple explanation? Thanks in advance.

A certain square is to be drawn on a coordinate plane. One of the vertices must be on the origin, and the square is to have an area of 100. If all coordinates of the vertices must be integers, how many different ways can this square be drawn?

i would like to know too... I tried to draw it out besides the 4 and i could draw some diamonds on the 4 directions but to get a side of 10 on the square in the 1-1-sqrt(2) format the x and y need to be 5*sqrt(2) which isnt integer... my guess would be B...
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Area = 100 each side=10 as one vertex is at the origin, one vertex is a x intercept and another y intercept.the fourth can be on the four coordinates.
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Keep trying no matter how hard it seems, it will get easier.

certain square is to be drawn on a coordinate plane. One of the vertices must be on the origin, and the square is to have an area of 100. If all coordinates of the vertices must be integers, how many different ways can this square be drawn? 4 6 8 10 12

first + 1 for posting a good question !

IMO It should be e, We can 4 simple squares with 2 sides on the X and Y coordinates as shown ( 1 in each Quadrant ) And now Lets consider one side of the square as OA and try moving it. OA = 10, Lets consider OA as hypotenuse of a right angled triangle with one side X axis as one coordinate, A can have integers as it coordinates only when either x= 8 or 6 and Y as 6 or 8 .. as OA^2 = X^2 + Y^2 ( 10 x10 = 8x8 + 6x6) for any other combination A will no be having its coordinates as integers for eg if X =9 and then y = sqrt 19. So , This way we can have 2 squares in 1st quadrant and similarly total 8 in 4 quadrants.

certain square is to be drawn on a coordinate plane. One of the vertices must be on the origin, and the square is to have an area of 100. If all coordinates of the vertices must be integers, how many different ways can this square be drawn? 4 6 8 10 12

first + 1 for posting a good question !

IMO It should be e, We can 4 simple squares with 2 sides on the X and Y coordinates as shown ( 1 in each Quadrant ) And now Lets consider one side of the square as OA and try moving it. OA = 10, Lets consider OA as hypotenuse of a right angled triangle with one side X axis as one coordinate, A can have integers as it coordinates only when either x= 8 or 6 and Y as 6 or 8 .. as OA^2 = X^2 + Y^2 ( 10 x10 = 8x8 + 6x6) for any other combination A will no be having its coordinates as integers for eg if X =9 and then y = sqrt 19. So , This way we can have 2 squares in 1st quadrant and similarly total 8 in 4 quadrants.

Hence total no. of squares will 8 + 4 =12.

Let me know if this is correct. ! Cheers

Each side of the square must have a length of 10. If each side were to be 6, 7, 8, or most other numbers, there could only be four possible squares drawn, because each side, in order to have integer coordinates, would have to be drawn on the x- or y-axis. What makes a length of 10 different is that it could be the hypotenuse of a Pythagorean triple, meaning the vertices could have integer coordinates without lying on the x- or y-axis.

For example, a square could be drawn with the coordinates (0,0), (6,8), (-2, 14) and (-8, 6). (It is tedious and unnecessary to figure out all four coordinates for each square).

If we label the square abcd, with a at the origin and the letters representing points in a clockwise direction, we can get the number of possible squares by figuring out the number of unique ways ab can be drawn.

a has coordinates (0,0) and b could have the following coordinates:

A certain square is to be drawn on a coordinate plane. One of the vertices must be on the origin, and the square is to have an area of 100. If all coordinates of the vertices must be integers, how many different ways can this square be drawn?

(A) 4 (B) 6 (C) 8 (D) 10 (E) 12

i think the question boiles down to the following (a,b) is a coordinate pair that must satisfy the following criteria

intigers as well as a^2+b^2 = 100

(10,10),(-10,10),(10,-10),(-10,-10) (8,6),(6,8),(-8,-6),(-6,-8),(-8,6),(6,-8),(-6,8),(8,-6) total number = 12

E, so sides must be 10, with either x, y to be a +-10 and 0 pair, or x^2+y^2 = 10^2, so x could be +-6, +-8, and y could be +- 8, +- 6 so that's 12 different starting lines, 12 different squares

A certain square is to be drawn on a coordinate plane. One of the vertices must be on the origin, and the square is to have an area of 100. If all coordinates of the vertices must be integers, how many different ways can this square be drawn? 1. 4 2. 6 3. 8 4. 10 5. 12

OA to come this afternoon

Well I think they are only 4 ways so the answer is A (or 1. in this case)

it says that there must be one vertice on the origin so you can draw a square in each of the cuadrants having 4 in total.

We can draw 12 squares with one vertix at the origin. The regular 4 squares with all vertices on the x and y coordinates - 4 squares Tilt the side of a square to 45 degrees - 4 squares Tilt the side of the square to 60 degrees - 4 squares

Hmm... thanks. I didn't think of tilting the squares.. +1
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We can draw 12 squares with one vertix at the origin.

Since \sqrt{100} = \sqrt{36 + 64}

we can have 2 points with distance 10 from origin. (8,6) and (6,8). Also not to forget (10,0). So in all 3 valid points in first quadrant. As we rotate this square in each quadrant we will have 3*4 = 12

So the answer is 12.
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Thanks, Just think differently, there is a easier solution:)

We can draw 12 squares with one vertix at the origin.

Since \sqrt{100} = \sqrt{36 + 64}

we can have 2 points with distance 10 from origin. (8,6) and (6,8). Also not to forget (10,0). So in all 3 valid points in first quadrant. As we rotate this square in each quadrant we will have 3*4 = 12

maybe this is to simple, but if its a square than all sides have to be the same. And if there are 4 vertices of a square and you cant change the size of the square than wouldnt there be four options? Drawing a square in each of the 4 quadrants?

A certain square is to be drawn on a coordinate plane. One of the vertices must be on the origin, and the square is to have an area of 100. If all coordinates of the vertices must be integers, how many different ways can this square be drawn? 4 6 8 10 12

In short, this question is asking how many combinations of integers x & y that form the relationship (x)squared * (y)squared = 100 they are (0,10),(6,8) and (8,6) Ignoring the other vertices, this shows the 3 options of one vertix in a single quadrant. Now multiply this by the number of quadrants i.e. 4 You get the number of ways of drawing the square will be 3*4 = 12!

I've been having a hard time understanding the solution to this problem. Can anyone offer a simple explanation? Thanks in advance.

A certain square is to be drawn on a coordinate plane. One of the vertices must be on the origin, and the square is to have an area of 100. If all coordinates of the vertices must be integers, how many different ways can this square be drawn?

***The Answer is 12 but I don't understand why.

one of the vertex is origin so (0,0) now area is 100 to get 100 the length of the side can be 10, the set of coordinates on x and y plane dat can have hypotenuse as 10 from origin (0,10), (10,0), (0,-10), (-10, 0) (6,8),(-6,-8), (-6,8), (6,-8), (8,6), (-8,6), (8,-6), (-8,-6),

Hi all. Taking the GMATs tomorrow. Needed some help for the day. Geometry:

stmt1: QR = RS, => <RQS = <RSQ not suff to find angle x stmt2: ST = TU, => <TSU = <TUS not suff to find angle x

if we use both in quad PQSU, => <p+<q+<s+<u = 360 => 90+180 - <RQS+x+180 - <SUT = 360 =>90+x-<RSQ-<TSU = 0 since <RQS = <RSQ and <TSU = <TUS but x+<RSQ+TSU = 180 => <RSU+<TSU = 180 - x replace this in above equation 90+x-180+x = 0 => 2x-90=0 => x is 45

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