If ABCD is a square, what are the coordinates of C?

Question

squarecoordinates_figure.PNG If ABCD is a square, what are the coordinates of C? A. \sqrt{3} , \sqrt{3} B. \sqrt{3}, 1 + \sqrt{3} C. 2 \sqrt{3} , \sqrt{3} D. 1 + \sqrt{3} , \sqrt{3} E. \sqrt{3} , 2 \sqrt{3} Guys - I got the right answer. Please see my attachment in the 2nd file. I have a question though. In my 2nd attachment (Answer.pdf) why can't angle C be 60 degrees and D be 30 degrees?

PareshGmat · Accepted Answer

Please refer the screenshot

It is a thumb rule; if a right triangle has angles 30 - 60 - 90; sides correspondingly opposite would be  1 - \sqrt{3} - 2

Bunuel · Answer

That's because the straight line is 180 degrees (30+90+60) and not 150 degrees (30+90+30).

PareshGmat · Answer

In this example, all the right triangles formed would be 30 - 60 - 90 as per diagram below;

GeorgeA023 · Answer

If Side =2 then the diagonal equals  2\sqrt{2}

The diagonal equals the distance between the 2 points. So I assume we can use the distance between 2 points formula.

D= \sqrt{(x2-x1)^2+(y2-y1)^2} 
 D^2 = (x2-x1)^2+(y2-y1)^2

2\sqrt{2}^2 = (x2-0)^2+(y2-1)^2

2\sqrt{2}^2 = (1+\sqrt{3})^2+(\sqrt{3}-1)^2

8= (1+\sqrt{3})^2+(\sqrt{3}-1)^2

8=4+2

???

Can anyone help and tell me where I went wrong.

I thought I could apply the distance between 2 points formula because we knew the distance, and points X1 and Y1.

GeorgeA023 · Answer

If anyone can help please do so. I did some practice problems using the formula and it works, however, it does not work for this problem. I emailed Magoosh, where the Q comes from, but they never respond.

KarishmaB · Answer

You can. Everything is correct except the last calculation.

8= (1+\sqrt{3})^2+(\sqrt{3}-1)^2

8= 1^2 + \sqrt{3}^2 + 2*1*\sqrt{3} + \sqrt{3}^2 + 1^2 - 2*\sqrt{3}*1 
 8 = 1 + 3 + 3 + 1 = 8

KarishmaB · Answer

There are many other ways of solving this question:

One method is to note that the sides of 30-60-90 triangle are in the ratio 1: \sqrt{3} :2. Since OA has length 1, OD has length  \sqrt{3} . Because the square is tilted, x coordinate of C will be more than x coordinate of D. So options (A), (B) and (E) are out. Also, the square is slightly titled so the x coordinate of C will not be twice of x coordinate of D. Hence option (C) is not possible either. Answer must be (D).

We see that coordinates of D are ( \sqrt{3} , 0) because of the 30-60-90 triangle. Now imagine that the x axis is turned 90 degrees at D. So the y coordinate of C will be  \sqrt{3} . Also since OA is of length 1, when you rotate it by 90 degrees, the x coordinate of C is 1 more than the x coordinate of C i.e. it is  \sqrt{3} + 1 .

unceldolan · Answer

Is this a special pythagorean triangle like the ones with sides 3,4,5 and 5,12,13?

KarishmaB · Answer

Pythagorean triplets give you the ratio of sides where all three lengths of sides are integers - the most common one is 3, 4, 5.

This is the ratio of sides in case the right triangles are special triangles:

Triangle with angles 30-60-90 (occur often) - Here ratio of sides is 1: root 3: 2 (Not a pythagorean triplet since all sides are not integers)
Isosceles right triangle i.e. angles are 45-45-90 - Here ratio of sides is 1:1:root 2 (Not a pythagorean triplet since all sides are not integers)

robinpallickal · Answer

Can anyone Pls. explain , how did we get 1 for one of the sides of the blue triangle. I solved till that point and could not proceed.

ENGRTOMBA2018 · Answer

As mentioned above, after you get the angles as 30-60-90 in the blue triangle, you need to remember that the sides for a 30-60-90 triangle are in the ratio  1:\sqrt{3}:2  with '1' side opposite to 30 degree angle. Thus you get 1 for that length.

Another way is by trigonometry: in the blue triangle, you know that the hypotenuse =2

Thus in this triangle, cos (60) = base / hypotenuse ---> 1/2 = base / 2 ---> base = 1 (as cos(60) = 0.5)

GMATinsight · Answer

An explanation using property of slopes!

Product of the slopes of Perpendicular lines = -1

m_1*m_2 = -1 
Since AO = 1 therefore OD = √3  
Let, Slope of Line AD,  m_1 = -1/√3 
i.e. Slope of Line DC,  m_2 = √3 
in Triangle i.e. Ratio of Y-Co-ordinate of C to Horizontal Distance of CD on X-Axis = √3/1

i.e. x Co-ordinate of C = OD+1= √3+1

i.e. Co-ordinates of C = ((√3+1),√3)

i.e. Option D

mvictor · Answer

since we have 30-60-90 triangle, we can deduct that the sides of the small triangle with the base on the x axis is 1-sqrt(3)-2
which makes the length of a side of the square = 2.
since all the sides are equal, we can draw a line from point c to the x-axis to form a 30-60-90 triangle.
since we have a similar triangle with the first one, we see that the hypotenuse is the same = 2. which makes that the point on y axis is sqrt(3)

now, only points A, C, and D have y coordinate of point c sqrt(3). B and E are thus eliminated.
now, from the origin to the point D, we have a distance of sqrt(3). The distance from point D to the line perpendicular to the x-axis which form a 30-60-90 triangle is 1. Thus, the x-coordinate of the point C must be 1+sqrt(3)

Eliminate A, and C.

D is the answer.

email2vm · Answer

For Red Triangle
cos 60 =adjancent side/hyp
1/2 =1/h
h=2

cos30=\sqrt{3}/2=x/2
x axis length = \sqrt{3}

now the blue side
cos 60=1/2=x/2
x=1

so total lenght of the x axis is 1+\sqrt{3}

only D is the suitable option

beyyaar · Answer

VeritasKarishma

How distance formula helped find the coordinate?? I did not get this method.
Kindly help.

beyyaar · Answer

VeritasKarishma

I believe it's just trying out options and getting 8 as distance?? 
Will that not take time? Or am I missing something here??

KarishmaB · Answer

Yes, it is taking options and plugging them in to find the one that yields a diagonal length of  2\sqrt{2} . Personally, I don't favour this method. Check out my reply above to see how I would solve this question.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If ABCD is a square, what are the coordinates of C?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If ABCD is a square, what are the coordinates of C?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards