What is the length of segment BC?

Question

https://gmatclub.com/forum/download/file.php?id=6447 What is the length of segment BC? (1) Angle ABC is 90 degrees. (2) The area of the triangle is 30. Triangle.jpg

Bunuel · Accepted Answer

Responding to a pm.

What is the length of segment BC?

(1) Angle ABC is 90 degrees.

From this, we can derive that \(BC=\sqrt{13^2-5^2}\). Sufficient.

(2) The area of the triangle is 30.

Consider the diagram below:

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 \(area=\frac{1}{2}baseheight=\frac{1}{2}13\frac{60}{13}=30\). So, as you can see, we can have different lengths of segment CB (\(CB_1\) and \(CB_2\)). Not sufficient.

Answer: A.

Hope it's clear.

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Attachment:

ABC.png [ 6.06 KiB | Viewed 39624 times ]

walker · Answer

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.

mymba99 · Answer

What a bliss! I was being jerk!
For those who did not understand Walker's expln, check out this link.
(Interactive triangle)
https://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 :-D

Attachments

traingle.doc [39.5 KiB]
Downloaded 673 times

hoping_for_stern · Answer

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

himang · Answer

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

Bunuel · Answer

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.

gmatkum · Answer

(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 = √  or A² =   
(30)² =   *   *   *   
900 =   *   *   *   
900 =    
Let (x/2)² = y 
900 =    
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.

AliciaSierra · Answer

Bunuel , why can't we find third side of triangle using Heron's formula i.e. https://www.mathopenref.com/heronsformula.html

Bunuel · Answer

Because Heron's Formula will give two positive values for the third side.

Heron's Formula:  Area = \sqrt{p(p-a)(p-b)(p-c)} , where a, b, and c are the lengths of the sides of a triangle and p is half the perimeter.

Substituting values from the question we get:  30=\sqrt{\frac{18+x}{2}(\frac{18+x}{2}-13)(\frac{18+x}{2}-5)(\frac{18+x}{2}-x)} , where x is the length of the third side. This gives two positive values for x:  x = 12  and  x=2\sqrt{61} . So, (2) is not sufficient.

Hope it's clear.

KartikSingh09 · Answer

Bunuel , I have seen this in a lot of questions where we interpret the figure differently than it is drawn. I know that unless angles are shown/given we are not sure about the size of the triangle but are we not sure about the shape as well?

There can be different ways to draw a certain triangle/quadrilateral. How do we know when to take the diagram at its face value and when to interpret it as a different shape/size?

Thanks,
Kartik

Bunuel · Answer

Check the instructions you get before the exam:
 https://gmatclub.com/forum/download/file.php?id=66457

https://gmatclub.com/forum/download/file.php?id=66458

https://gmatclub.com/forum/download/file.php?id=66459

I'd advice to familiarize yourselves with the above, especially pay attention to the parts in red boxes.

Here is a part you are interested in:

For  all  questions in the Quantitative section you may assume the following: 
  Numbers:  
  All numbers used are real numbers.

Figures:  
  For Problem Solving questions, figures are drawn as accurately as possible. Exceptions will be clearly noted.  
   For Data Sufficiency questions, figures conform to the information given in the question, but will not necessarily conform to the additional information given in statements (1) and (2).   
  Lines shown as straight are straight, and lines that appear jagged are also straight.  
  The positions of points, angles, regions, etc. exist in the positing shown, and angle measures are greater than zero. 
  All figures lie in a plane unless otherwise indicated.

Hope it helps.

Shrihari12 · Answer

Hi  Bunuel ,

in St. 2 can we also say that as we do not know which Angle is 90Degree so we cannot conclude which side can be the Height? And thus we cannot say for certain the height of the triangle? Thus Statement 2 will be insufficient?

bumpbot · Answer

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What is the length of segment BC?

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