In the right triangle above, a

Question

https://gmatclub.com/forum/download/file.php?id=35644 In the right triangle above, a < b. What is a? (1) c = 5 (2) The area of the triangle is 6. RightTriangle.png

nishantt7 · Answer

I think the answer should be A.
Statement 1 is sufficient becoz c=5 gives a=3 and b=4. The only possible solution.
Statement 2 gives a=3 and b=4 or a=2 and b=6.

naorba · Answer

Testing 1st statement:

even if were given that c = 5, it does not necessarily mean that the triangle is of the form   3:4:5  , since there could be more options:
Pythagorean theorem --->  a^2+b^2=c^2 
 1:\sqrt{24}:5  ------>  1^2+\sqrt{24}^2 = 25  Hence,  not sufficient

Testing 2nd statement:

Triangle Area = \frac{a*b}{2} , so:
 \frac{a*b}{2} = 6  --- > multiplying by 2
 a*b = 12 
still, not sufficient

1st+2nd statement combined:

Factors of 12:
1:12
2:6
 3:4

only combination that fits with the Pythagorean theorem is  3:4 :5

Hence, Answer is (C)

0akshay0 · Answer

for St 1 what if a = root(5) and b = 2root(5) (since nothing is said about a or b) this will also satisfy a<b and the Pythagoras theorem.

alexjst · Answer

It does not say that the sides are integers, therefore we can not find a unique value knowing the hypotenuse and the area. I'd go with E

naorba · Answer

In order to say that C in not true you need to show that there is at list one more option for values that can fit the 2 statements above: area = 12, and c=5

Mo2men · Answer

Fact 1: c =5

No info about b.

Insufficient

(2) The area of the triangle is 6.

1/2 ab =6.......ab =12......No certain info about b.

Insufficient

Combine 1 & 2

For right triangle rule:  c^{2}  =  a^{2}  +  b^{2}

Multiply the equation in  b^{2}

25 b^{2} =  a^{2}  b^{2}  +  b^{4} ................Equation A

From fact 2: ab = 12...... (ab)^{2} = 144........Substitute in equation A

25 b^{2} = 144 +  b^{4}

b^{4} - 25 b^{2}  + 144 = 0

( b^{2}  - 9) ( b^{2}  -16) = 0

b^{2}  = 9 or  b^{2} = 16

Case 1: 
 b = 3 
 a =4
 Rejected as a < b

Case 2:

b = 4 
 a =3

Accepted as as a < b

One solution obtained

Answer: C

akshayk · Answer

S1 -> Insufficient. a A and B can take any values that satisfies the pythagorean theorem. S2 -> Insufficient. (ab)/2 = 6 => ab = 12 Multiple possible values. S1+ S2 -> Sufficient a a is 3, b is 4. The answer is C.

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In the right triangle above, a < b. What is a?

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