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What is the length of segment AC? [#permalink]
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07 Mar 2018, 12:28
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In the figure above, what is the length of segment AC (not drawn)?
I. AB = BC
II. BC = 4Source: ExamPAL free preparation videos. It has been beautifully explained in their prep video. Posting here for everyone to try.
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Last edited by chetan2u on 16 Mar 2018, 05:48, edited 1 time in total.
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Re: What is the length of segment AC? [#permalink]
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07 Mar 2018, 14:13
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Answer choice should be A.
You can find out the angle B from the property of a quadrilateral [sum of all angles 360 deg]. So B = 60. Join AC. Now with 1,
AB = BC, so ABC is equilateral. [Base angles equal and vertical angle 60]. Now the next coming to ADC.
If you subtract the base angles of the adjacent triangle (which you just computed to be 60 deg), you will see that the triangle is actually a 459045 triangle. Hence you can say AD = AB= BC = AC = 4.



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Re: What is the length of segment AC? [#permalink]
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08 Mar 2018, 03:49
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srinjoy1990 wrote: Answer choice should be A.
You can find out the angle B from the property of a quadrilateral [sum of all angles 360 deg]. So B = 60. Join AC. Now with 1,
AB = BC, so ABC is equilateral. [Base angles equal and vertical angle 60]. Now the next coming to ADC.
If you subtract the base angles of the adjacent triangle (which you just computed to be 60 deg), you will see that the triangle is actually a 459045 triangle. Hence you can say AD = AB= BC = AC = 4. How is this equilateral triangle ? Bunuel #Bunuel



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Re: What is the length of segment AC? [#permalink]
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08 Mar 2018, 04:11
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shringi87 wrote: How is this equilateral triangle ? Hey shringi87 , Please note that sum of all angles of a quadrilateral is 360. That means angle B is 360  angle A  Angle C  angle D = 360  150  45  105 = 60. Therefore, we have angle B = 60. Now, join AC. In Triangle ABC, Angle B = 60 . Using statement A, we have AB = BC. This means angle BAC = angle BCA (isosceles triangle). Also, angle BAC + angle BCA + angle ABC = 180 => Angle BAC + angle BCA + 60 = 180 => angle BAC = 120/2 = 60. Since we got all angles = 60, we can say Triangle ABC is an equilateral triangle. Does that make sense?
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Re: What is the length of segment AC? [#permalink]
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08 Mar 2018, 09:50
Why (B) insufficient ?



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Re: What is the length of segment AC? [#permalink]
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08 Mar 2018, 09:53
AlexGmat2017 wrote: Why (B) insufficient ? Hey AlexGmat2017 , B is insufficient because it gives the length of only one side. Although we can find out angle B, it may happen that BC is not equal to AB. Hence, we cannot find out AC. Does that make sense?
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Re: What is the length of segment AC? [#permalink]
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08 Mar 2018, 11:54
Pls expand on. If we know all angles, AD (from figure) and BC (as given from st. 2) then our quadrilateral is fixed. If you think no, why?
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Re: What is the length of segment AC? [#permalink]
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09 Mar 2018, 00:25
I chose (C) but I dont understand why it is wrong and how (A) is correct. While we can see that ABC is a equilateral triangle, how do we get the exact value of the lengths without any reference provided (such as found in statement 2?) any kind souls that can help me?



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Re: What is the length of segment AC? [#permalink]
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09 Mar 2018, 02:05
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AlexGmat2017 wrote: Pls expand on. If we know all angles, AD (from figure) and BC (as given from st. 2) then our quadrilateral is fixed. If you think no, why? Hey AlexGmat2017 , How can you say that the quadrilateral is fixed when two opposite sides are given (AD in the question and BC from statement 2.) ? I can make AD vertical keeping the length = 4 and still make my quadrilateral. Hence, Statement 2 is not sufficient to determine AC. Feel free to reach out in case of any concern. ryanwxj wrote: I chose (C) but I dont understand why it is wrong and how (A) is correct. While we can see that ABC is a equilateral triangle, how do we get the exact value of the lengths without any reference provided (such as found in statement 2?) any kind souls that can help me? Hey ryanwxj , I am not sure whether my soul is kind but I would like to still help you. We have one reference provided. Look at the figure carefully. We have AD = 4. Since we know that ABC is equilateral, we can very well determine the angles of ADC. They will be 90 and 45. So, basically we are getting a 456090 triangle. Hence, using the property of such triangles, we can determine the length of AC. Does that make sense?
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Re: What is the length of segment AC? [#permalink]
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09 Mar 2018, 02:21
abhimahna wrote: AlexGmat2017 wrote: Pls expand on. If we know all angles, AD (from figure) and BC (as given from st. 2) then our quadrilateral is fixed. If you think no, why? Hey AlexGmat2017 , How can you say that the quadrilateral is fixed when two opposite sides are given (AD in the question and BC from statement 2.) ? I can make AD vertical keeping the length = 4 and still make my quadrilateral. Hence, Statement 2 is not sufficient to determine AC. Feel free to reach out in case of any concern. ryanwxj wrote: I chose (C) but I dont understand why it is wrong and how (A) is correct. While we can see that ABC is a equilateral triangle, how do we get the exact value of the lengths without any reference provided (such as found in statement 2?) any kind souls that can help me? Hey ryanwxj , I am not sure whether my soul is kind but I would like to still help you. We have one reference provided. Look at the figure carefully. We have AD = 4. Since we know that ABC is equilateral, we can very well determine the angles of ADC. They will be 90 and 45. So, basically we are getting a 456090 triangle. Hence, using the property of such triangles, we can determine the length of AC. Does that make sense? Oh gosh! How could I have missed it Yes you are right, with AD known, it makes a lot of sense now. Thanks so much! And you are indeed a very kind soul



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Re: What is the length of segment AC? [#permalink]
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09 Mar 2018, 05:51
abhimahna wrote: AlexGmat2017 wrote: Pls expand on. If we know all angles, AD (from figure) and BC (as given from st. 2) then our quadrilateral is fixed. If you think no, why? Hey AlexGmat2017 , How can you say that the quadrilateral is fixed when two opposite sides are given (AD in the question and BC from statement 2.) ? I can make AD vertical keeping the length = 4 and still make my quadrilateral. You cannot make AD vertical bc angle ADC=45. Posted from my mobile device



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Re: What is the length of segment AC? [#permalink]
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09 Mar 2018, 06:12
AlexGmat2017 wrote: You cannot make AD vertical bc angle ADC=45. Yes, that's correct. But I can move AD to the right as well as left keeping all angles and AD length same, right? This would change the length of AC.
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Re: What is the length of segment AC? [#permalink]
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09 Mar 2018, 09:22
abhimahna wrote: AlexGmat2017 wrote: You cannot make AD vertical bc angle ADC=45. Yes, that's correct. But I can move AD to the right as well as left keeping all angles and AD length same, right? This would change the length of AC. No, you cannot move AD the left or to the right bc its AD=4. Posted from my mobile device



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Re: What is the length of segment AC? [#permalink]
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09 Mar 2018, 09:34
AlexGmat2017 wrote: No, you cannot move AD the left or to the right bc its AD=4.
Hey AlexGmat2017 , I think you are considering the figure drawn to scale. Remember if we are not given "Figure is drawn to scale" , you cannot assume that. Thanks
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Re: What is the length of segment AC? [#permalink]
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09 Mar 2018, 11:08
abhimahna wrote: AlexGmat2017 wrote: No, you cannot move AD the left or to the right bc its AD=4.
Hey AlexGmat2017 , I think you are considering the figure drawn to scale. Remember if we are not given "Figure is drawn to scale" , you cannot assume that. Thanks Please, one more general question for you. If you know all angels in quadrilateral and any two of four its side segments (assuming no two side segments are parallel) can you specify this quadrilateral unambiguously?



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Re: What is the length of segment AC? [#permalink]
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09 Mar 2018, 14:33
ryanwxj wrote: I chose (C) but I dont understand why it is wrong and how (A) is correct. While we can see that ABC is a equilateral triangle, how do we get the exact value of the lengths without any reference provided (such as found in statement 2?) any kind souls that can help me? Exactly where I went wrong.. and exactly what I thought first ... there is a small 4 drawn in the figure showing the side AD. Without that  (C) would be the correct answer.
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Re: What is the length of segment AC? [#permalink]
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10 Mar 2018, 22:11
Jiggy11 wrote: In the figure above, what is the length of segment AC (not drawn)?
I. AB = BC
II. BC = 4Source: ExamPAL free preparation videos. It has been beautifully explained in their prep video. Posting here for everyone to try.I would disagree with OA and feel the OA was decided without giving sufficient thought to the meaning of the statement II.. statement I...AB=BC see the attached figure trape1[img]trape1jpg[/img] we can work out angle B as 60 ... 36045150105=60 Ab = BC means Angle ACB = BAC = 120/2=60, so ABC is an equilateral triangle.. angle DAC = 15060=90 and angle DCA = 10560=45 so ACD is isosceles right angled triangle thus AD = AC = 4 suff statement IIBC=4 draw the quadrilateral.. see att fig 2#1. draw a line with one end at D.. #2 At D draw 45 angle. #3 Cut 4 on this line and name that point A #4 at A draw 150 angle Now these two lines AB and CD are not parallel so would meet each other on LEFT side of AD.. If you draw paralle line at an angle 105 on this base line (line starting with A), the length of BC will increase from 0 at which the lines intersect and move to infinity as it moves right.. Nowhere the length of BC can be same at any two location, so BC =4 will be a unique point and thus we will have a unique quadrilateral... When you have a unique quadrilateral, all sides and diagonals can have ONLY one value, so Ac can be measured.. Sufficient NOTE.. HAd the two lines been parallel then BC would be 4 at all points and AC could have various values D
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Re: What is the length of segment AC? [#permalink]
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16 Mar 2018, 03:37
chetan2u wrote: Jiggy11 wrote: In the figure above, what is the length of segment AC (not drawn)?
I. AB = BC
II. BC = 4Source: ExamPAL free preparation videos. It has been beautifully explained in their prep video. Posting here for everyone to try.I would disagree with OA and feel the OA was decided without giving sufficient thought to the meaning of the statement II.. statement I...AB=BC see the attached figure trape1[img]trape1jpg[/img] we can work out angle B as 60 ... 36045150105=60 Ab = BC means Angle ACB = BAC = 120/2=60, so ABC is an equilateral triangle.. angle DAC = 15060=90 and angle DCA = 10560=45 so ACD is isosceles right angled triangle thus AD = AC = 4 suff statement IIBC=4 draw the quadrilateral.. see att fig 2#1. draw a line with one end at D.. #2 At D draw 45 angle. #3 Cut 4 on this line and name that point A #4 at A draw 150 angle Now these two lines AB and CD are not parallel so would meet each other on LEFT side of AD.. If you draw paralle line at an angle 105 on this base line (line starting with A), the length of BC will increase from 0 at which the lines intersect and move to infinity as it moves right.. Nowhere the length of BC can be same at any two location, so BC =4 will be a unique point and thus we will have a unique quadrilateral... When you have a unique quadrilateral, all sides and diagonals can have ONLY one value, so Ac can be measured.. Sufficient NOTE.. HAd the two lines been parallel then BC would be 4 at all points and AC could have various values D Hi Chetan2u So is the right answer D (as you say and I would tend to follow you on that) or A as it is still written at the top? Thanks




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