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If the vertices of a triangle have coordinates \((x, 1)\), \((5, 1)\), and \((5, y)\) where \(x \lt 5\) and \(y \gt 1\), what is the area of the triangle? (1) \(x = y\) (2) Angle at the vertex \((x, 1)\) is equal to angle at the vertex \((5, y)\)
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16 Sep 2014, 00:36
Official Solution: Look at the diagram below: Notice that vertex (x,1) will be somewhere on the green line segment and the vertex (5,y) will be somewhere on the blue line segment. So, in any case our triangle will be right angled, with a right angle at vertex (5, 1). Next, the length of the leg on the green line segment will be \(5x\) and the length of the leg on the blue line segment will by \(y1\). So, the area of the triangle will be: \(\text{area}=\frac{1}{2}*(5x)*(y1)\). (1) \(x=y\). Since \(x \lt 5\) and \(y \gt 1\) then both \(x\) and \(y\) are in the range (1,5): \(1 \lt (x=y) \lt 5\). If we substitute \(y\) with \(x\) we'll get: \(\text{area}=\frac{1}{2}*(5x)*(y1)=\frac{1}{2}*(5x)*(x1)\), different values of \(x\) give different values for the area (even knowing that \(1 \lt x \lt 5\)). Not sufficient. (2) Angle at the vertex \((x,1)\) is equal to angle at the vertex \((5,y)\). We have an isosceles right triangle: \(5x=y1\). Again if we substitute \(y1\) with \(5x\) we'll get: \(\text{area}=\frac{1}{2}*(5x)*(y1)=\frac{1}{2}*(5x)*(5x)\), different values of \(x\) give different values for the area. Not sufficient. (1)+(2) \(x=y\) and \(5x=y1\). Solve for \(x\): \(x=y=3\), so \(\text{area}=\frac{1}{2}*(53)*(31)=2\). Sufficient. Answer: C
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Re: M0806
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31 Dec 2015, 11:16
Hello Bunuel, I have a concern regarding this question. I think OA should be E not C. How can x=y translates into 5−x=y−1 ?? Please see the attached pic, using which we can see that even when X = Y areas could be different. Please assist. Thanks
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Re: M0806
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12 Jan 2016, 23:34
WillGetIt wrote: Hello Bunuel,
I have a concern regarding this question. I think OA should be E not C.
How can x=y translates into 5−x=y−1 ??
Please see the attached pic, using which we can see that even when X = Y areas could be different.
Please assist.
Thanks Hi WillGetIt, With both conditions being true, you will get only a single isosceles triangle ((3,1), (5,1), (5,3)). Only these points will satisfy both requirements. Hope this helps. Regards, Renin



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Re: M0806
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13 Jan 2016, 04:48
Already explained well. For the area, we need both height and base. So, here it is only possible, if we can know all vertices. Can you get the answer using (1) only, no, since many possibilities such as, (2,1),(5,2);(3,1),(5,3);(4,1),(5,4)..etc.Insufficient Forget about (1), think on (2), but still many possibilities when equal angles give us equal sides,such as (4,1),(5,2);(3,1),(5,3);(1,1),(5,1)..etc, Insufficient. So, A,B,D all cancelled since not at least one sufficiency. Now combine both, ...(2) requires that both distance from 5 will be equal. Make it narrower using (1)..so that finally x=y. Only one case is possible here which is (3,1),(5,3). Because we can calculate the area combinedly, C is the answer.



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14 Jul 2016, 11:28
I think this is a highquality question and I agree with explanation. Great question. Any similar coordinate geometry questions for practice outside GMATCLUB tests please suggest such questions are awesome man!



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29 Aug 2016, 21:44
I'm guessing we can't assume x=y to be an equation of a line? Can you confirm? Tx.



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30 Aug 2016, 02:31



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M0806
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Updated on: 09 Oct 2016, 20:43
This just clicked. Thank you.
Originally posted by wmichaelxie on 30 Aug 2016, 07:02.
Last edited by wmichaelxie on 09 Oct 2016, 20:43, edited 1 time in total.



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Re: M0806
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02 Sep 2016, 00:22
Bunuel wrote: If the vertices of a triangle have coordinates \((x, 1)\), \((5, 1)\), and \((5, y)\) where \(x \lt 5\) and \(y \gt 1\), what is the area of the triangle?
(1) \(x = y\)
(2) Angle at the vertex \((x, 1)\) is equal to angle at the vertex \((5, y)\) Responding to a pm: Angle at (5, 1) will be 90 degrees. (1) \(x = y\) The coordinates are (x, 1), (5, 1) and (5, x). Not sufficient alone since we don't know x. (2) Angle at the vertex \((x, 1)\) is equal to angle at the vertex \((5, y)\)[/quote] This means the two sides are equal. So 5  x = y  1 Not sufficient alone. Using both statements, 5  x = x  1 x = 3 = y We get all coordinates so we will be able to find the area. Answer (C)
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Re: M0806
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12 Oct 2016, 09:39
I am able to understand line distance as y1 is because of sqrt((55)^2+(y1)^2) = y1 but, why the other distance is 5x and not x5 as the calculation would be sqrt((x5)^2/+(11)^2) = x5
can you please explain this.
Ashish



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Re: M0806
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08 Nov 2016, 00:42
hello, as per the graph, we can obtain all three vertices of the triangle with statement 1 itself. please tell me where am i thinking wrong?
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Re: M0806
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07 Jan 2017, 11:24
Sumanth8492, as per statement 1, vertices of x and y can be 2,3, or 4. We can not obtain the area of the triangle unless we know exactly which one.



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27 Oct 2017, 10:50
In this case, from the problem statement itself we can deduce that x & y are nothing but the two sides of the right angled triangle. So, doesn't x = y itself imply that the triangle is iso. rt angled triangle?In that case we already get to know that the angles are equal. Please let me know if I am missing something/understanding is incorrect.
TIA.



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27 Oct 2017, 10:57



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27 Oct 2017, 11:04
My bad. Thank you for the prompt response.










