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SVP  Joined: 29 Aug 2007
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What is the length of segment BC?  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 49% (01:10) correct 51% (01:11) wrong based on 630 sessions

HideShow timer Statistics What is the length of segment BC?

(1) Angle ABC is 90 degrees.
(2) The area of the triangle is 30.

Attachment: Triangle.jpg [ 28.62 KiB | Viewed 31428 times ]

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Originally posted by GMAT TIGER on 08 May 2008, 21:28.
Last edited by Bunuel on 09 Oct 2019, 21:23, edited 3 times in total.
Edited the question and added the OA.
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Re: What is the length of segment BC?  [#permalink]

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4
Responding to a pm.

What is the length of segment BC? (1) Angle ABC is 90 degrees. $$BC=\sqrt{13^2-5^2}$$. Sufficient.

(2) The area of the triangle is 30. Consider the diagram below:
Attachment: ABC.png [ 6.06 KiB | Viewed 16140 times ]
Notice that segment $$AB_2$$ is the mirror reflection of segment $$AB_1$$ around the vertical line passing through point A. Now, if the height of triangles $$ACB_1$$ and $$ACB_2$$ is $$\frac{60}{13}$$, then the area of both triangles is $$area=\frac{1}{2}*base*height=\frac{1}{2}*13*\frac{60}{13}=30$$. So, as you can see we can have different lengths of segment CB ($$CB_1$$ and $$CB_2$$). Not sufficient.

Hope it's clear.
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Re: What is the length of segment BC?  [#permalink]

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1
A

1. Suff since it's a 5-12-13 triangle
2. Insuff because the height to side 5, which can be calculated to be 12 may not necessarily be the leg.
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GMAT 1: 750 Q50 V40 Re: What is the length of segment BC?  [#permalink]

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Hi all!

My 2 cents....

Let imagine AB and AC as sticks bounded in point A by a hinge. So, we can move AB relatively to AC and change angle BAC form 0 to 180. The area of ABC triangle will change form 0 (angle BAC=0) through maximum value 12*5*13=32.5 (angle BAC=90) to 0 (angle BAC=180). If you imagine that, It will be obvious that for 0<area<32.5 we will get two possible value of BC and for area=32.5 we will get only one value of BC. In our case area=30<32.5. Therefore, the second condition is insufficient.
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Re: What is the length of segment BC?  [#permalink]

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What a bliss! I was being jerk!
For those who did not understand Walker's expln, check out this link.
(Interactive triangle)
http://www.mathopenref.com/heronsformula.html
For there are 2 triangles with the area 30.
1: 5,13,12 (For acute angle)
2: 5,13,15.5... (Not 15.5) ( For obtuse angle)
see the attachment. Thank you all Attachments traingle.doc [39.5 KiB]

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Re: What is the length of segment BC?  [#permalink]

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Contradicting statement 2 :
We can keep the fact in mind that triangles between two parallel lines have same area.(draw a line ll to AC / BC's value will vary and still satisfy the area being constant at 30 ,as base and height in this case are constant (AC) and distance between ll lines remain the same , which in this case would be the height of the triangle)

Heron's formula makes one jump to a quick conclusion about statement 2 being sufficient but an important point is overlooked by the fact using it to find the value of BC would give some polynomial ( x3 or x4 or whatever ....) and that will lead to multiple values of x
(at least 2 in this case ).
Hence insufficient.

Statement 1 is sufficient
Hence A
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Re: What is the length of segment BC?  [#permalink]

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hoping_for_stern wrote:
Contradicting statement 2 :
We can keep the fact in mind that triangles between two parallel lines have same area.(draw a line ll to AC / BC's value will vary and still satisfy the area being constant at 30 ,as base and height in this case are constant (AC) and distance between ll lines remain the same , which in this case would be the height of the triangle)

Heron's formula makes one jump to a quick conclusion about statement 2 being sufficient but an important point is overlooked by the fact using it to find the value of BC would give some polynomial ( x3 or x4 or whatever ....) and that will lead to multiple values of x
(at least 2 in this case ).
Hence insufficient.

Statement 1 is sufficient
Hence A

Answer Should Be "D" since area is equals= 0.5*A*B*SinQ; so from here we can get the included angle between the given two sides and then use Cosine law to get the third side easily.....

+1 KUDDUS if I'm Right
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Re: What is the length of segment BC?  [#permalink]

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himang wrote:
hoping_for_stern wrote:
Contradicting statement 2 :
We can keep the fact in mind that triangles between two parallel lines have same area.(draw a line ll to AC / BC's value will vary and still satisfy the area being constant at 30 ,as base and height in this case are constant (AC) and distance between ll lines remain the same , which in this case would be the height of the triangle)

Heron's formula makes one jump to a quick conclusion about statement 2 being sufficient but an important point is overlooked by the fact using it to find the value of BC would give some polynomial ( x3 or x4 or whatever ....) and that will lead to multiple values of x
(at least 2 in this case ).
Hence insufficient.

Statement 1 is sufficient
Hence A

Answer Should Be "D" since area is equals= 0.5*A*B*SinQ; so from here we can get the included angle between the given two sides and then use Cosine law to get the third side easily.....

+1 KUDDUS if I'm Right

That's not correct. The answer to this question is A, not D.

The formula you posted is correct: $$Area=\frac{1}{2}BA*CA*sinA$$ --> $$sinA=\frac{12}{13}$$, but from this we cannot calculate angle $$A$$, as $$sin A=Sin (180-A)$$. Meaning that this won't give us ONLY one value for angle A, thus we won't have ONLY one value for CB --> A can be acute angle, making B right angle AND making CB equal to 12 OR A can be obtuse angle, making B acute angle AND making CB greater then 12.

Check here: what-is-the-length-of-segment-bc-63639-20.html#p1105437

Hope it helps.
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Re: What is the length of segment BC?  [#permalink]

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(1) Angle ABC is 90 degrees.
Since ABC is right angled at B, we can easily find the length of BC using Pythagoras Theorem; SUFFICIENT.

(2) The area of the triangle is 30.
Although Heron's formula is out of GMAT scope, but still let me clarify it here that why statement 2 is NOT sufficient.

Let us assume that BC = x
Then s = (5 + 13 + x)/2 = (18 + x)/2 = 9 + (x/2)
Now area of triangle, A = √[s(s - a)(s - b)(s - c)] or A² = [s(s - a)(s - b)(s - c)]
(30)² = [9 + (x/2)] * [9 + (x/2) - 5] * [9 + (x/2) - 13] * [9 + (x/2) - x]
900 = [9 + (x/2)] * [4 + (x/2)] * [-4 + (x/2)] * [9 - (x/2)]
900 = [81 - (x/2)²][(x/2)² - 16]
Let (x/2)² = y
900 = [81 - y][y - 16]
900 = 81y - y² + 16y - 1296
y² - 97y + 2196 = 0
y² - 36y - 61y + 2196 = 0
y(y - 36) - 61(y - 36) = 0
(y - 61)(y - 36) = 0
y = 36, 61
(x/2)² = 36, (x/2)² = 61, which clearly implies that we are getting 2 values of x, which means 2 values of BC. So, statement 2 is NOT sufficient.

The correct answer is A.

I hope that helps.
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Re: What is the length of segment BC?  [#permalink]

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We dont know whether it is a right angle triangle
Hence stmt 1- ab is 9 not suff as we dont know the length of the other side
Stmt 2 also insuff similarly
Combining we get
The triangle can be 4 4 9 or 9 9 4 but 4 4 9 is not possible since the 3rd side of the triangle cannot be greater than the sum of the other two sides

Hence C
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: What is the length of segment BC?  [#permalink]

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Hi kasturi72,

In Fact 1, we're told that angle ABC is a RIGHT ANGLE. With the two sides that we're given in the prompt (5 and 13), we CAN determine the length of the missing side (it's 12), so THAT information is SUFFICIENT to answer the question.

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GMAT 1: 710 Q48 V40 Re: What is the length of segment BC?  [#permalink]

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GMAT TIGER wrote:
Attachment:
Triangle.jpg
What is the length of segment BC?

(1) Angle ABC is 90 degrees.
(2) The area of the triangle is 30.

This is such a shrewd trap type. I hope to God I don't fall for it in the exam.

A alone is sufficient but when you come down to B you tend to use the information that you got from A. Also I was thinking how smart I am to already know the pythagorean triplet and Bam! _________________
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Re: What is the length of segment BC?  [#permalink]

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Hi, I believe the answer should be D. If we change the angle from 90 to any other angles the height changes and it can not keep the area 30 considering the length of 2 sides are 5 and 13.
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Re: What is the length of segment BC?  [#permalink]

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emonmisra wrote:
Hi, I believe the answer should be D. If we change the angle from 90 to any other angles the height changes and it can not keep the area 30 considering the length of 2 sides are 5 and 13.

Please check under the spoiler in original post, OA is not D, it's A. Please read the thread to see WHY it's not D.
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Re: What is the length of segment BC?  [#permalink]

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GMAT TIGER wrote: What is the length of segment BC?

(1) Angle ABC is 90 degrees.
(2) The area of the triangle is 30.

Attachment:
Triangle.jpg

(1) If it is right angle triangle, we can consider pythagorean triplet 5-12-13. Sufficient
(2) We dont know the height of the given triangle and therefore cannot find BC. Insufficient

A is correct Re: What is the length of segment BC?   [#permalink] 09 Oct 2019, 22:29
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