if a, b, c are the length of sides of a triangle, is

Question

GMATBusters’ Quant Quiz Question -3 
if a, b, c are the length of sides of a triangle, is the area of the triangle less than 22?

(1) a^2 + b^2≠ c^2
(2) a+ b < 13

GMATBusters · Answer

Official Explanation

St 1) a^2 + b^2≠ c^2
what inference can one get from this?
It simply says that angle between side a and b is not right angle. 
NOT sufficient.

St 2)a+ b < 13
Since, the question is asking whether the area of the triangle is less than 22, lets us calculate the max possible area
to get the max area the triangle must be an isosceles right-angled triangle 
let us take a+b = 13
hence there's = 1/2 * 6.5*6.5 = 21.125
Now, this is also <22
so, the area of the triangle must be <22
Sufficient.

Answer B

shameekv1989 · Answer

if a, b, c are the length of sides of a triangle, is the area of the triangle less than 22?

(1) a^2 + b^2≠ c^2
(2) a+ b < 13

Area of triangle = \(\frac{1}{2}*b*h\)

i) Triangle is not a right angled triangle => Clearly insufficient as it doesn't tell anything about the area

ii) a+b < 13

As the third side has to be smaller than the sum of the 2 sides => c < 13 as well

We need to determine if Area of triangle = \(\frac{1}{2}*b*h\) < 22 =>
Area of triangle = \(b*h\) < 44

The area would be maximum i.e. (Multiplication of the 2 sides would be maximum when the sides are equal in length)
thus, they would be equal when the sides are 6.5, 6.5 (Boundary limits)

6.5*6.5 = 42.25 < 44

Hence this should be sufficient to tell us that the area would be less than 44

Answer - B

monirhossain2nd · Answer

Ans E

Stat 1:
Clearly insufficient
No numerical value

Stat 2:
If we assume 6+7=13
In that case the area would be= (1/2). 6.7=21 >20
If we assume 3+10=13
In that case the area would be= (1/2). 3.10=15<20
Not sufficient

Combined :
Not Sufficient

Raxit85 · Answer

Area of triangle <= 22?

1) a^2+b^2≠c^2
It implies any of the following,
a^2+b^2<c^2
a^2+b^2>c^2
We do not clearly know the values of a and b here.
So, we cannot tell anything about the area of the triangular region. Hence, Insufficient

2) a + b < 13
If a+b < 13, the sum could be as less as 3 and as big as 12.
In case of a = 1, b=2 will give us an area lesser than 20.
a = 5, b=7 will give us an area greater than 20. Hence, Insufficient

1) + 2)
We still can't clearly tell anything about the area of the triangle as there are many values of a and b as per both statements. Hence, Insufficient.

Ans. E

Kinshook · Answer

Asked: if a, b, c are the length of sides of a triangle, is the area of the triangle less than 22?

(1) a^2 + b^2≠ c^2
The triangle is not a right angled triangle with sides a and b and hypotenuse c
But so sides are known
NOT SUFFICIENT

(2) a+ b < 13
Since 13>a+b>c
c<13
For maximum area, let us take side a=b~6.5 and angle between them to be as close to 90 as possible
Maximum Area = 1/2 * (6.5)^2 sin x =< 1/2 (6.5)^2 = 21.125 < 22
SUFFICIENT

IMO B

Danchar42 · Answer

if a, b, c are the length of sides of a triangle, is the area of the triangle less than 22?

(1) a^2 + b^2≠ c^2
(2) a+ b < 13

Area of triangle = \(\frac{1}{2}*b*h\)

i) Triangle is not a right angled triangle => Clearly insufficient as it doesn't tell anything about the area

ii) a+b < 13

As the third side has to be smaller than the sum of the 2 sides => c < 13 as well

We need to determine if Area of triangle = \(\frac{1}{2}*b*h\) < 22 =>
Area of triangle = \(b*h\) < 44

The area would be maximum i.e. (Multiplication of the 2 sides would be maximum when the sides are equal in length)
thus, they would be equal when the sides are 6.5, 6.5 (Boundary limits)

6.5*6.5 = 42.25 < 44

Hence this should be sufficient to tell us that the area would be less than 44

Answer - B

AnirudhaS · Answer

(1) All we know from this is c is not the hypotenuse of the triangle ABC, if it is a right angles triangle. Not sufficient.

(2) a+b<13
We know that the equilateral triangle has the largest area.
therefore, side can be lets say 6.4
So, area of the the equilateral triangle with side 6.4 =  \frac{\sqrt{3}}{4}*a^2  = 17.3 (approx) < 22

Answer: B

mohagar · Answer

using option A and B if we assume equilateral or a isoleces traingle we see that it cannot be greater than 21

uc26 · Answer

1) This doesn't help us with finding the area. Just tells us that the triangle is not a right angled triangle. So NOT SUFFICIENT.

2) Given that a+b<13. We know that for a given perimeter, equilateral triangle will have the greatest area. Even if we try to maximize the lengths of the sides a and b. Say a=6.5 and b=6.49999 and we assume the triangle to be an equilateral triangle, we will find that the area of the triangle cannot exceed 22. So SUFFICIENT.

Correct answer: B

neharaj0001 · Answer

(B) Statement 2 Alone Sufficient
If a,b and c are the sides of the triangle with a+b<13, the largest area of a triangle possible would be when a and b are perpendicular to one another and equal.
In such a scenario, the largest value possible for a and b both, such that a=b is a=b=slightly less than 13/2. This largest area in that case comes to slightly less than b × h/2 = a × b/2 (since the largest area occurs when a and b are perpendicular) = (13/2 × 13/2 ) / 2
= 169/8
In other words the total largest possible area is slightly less than 169/8 = 21.125 < 22
Therefore, with statement b alone, we can say that the area will be definitely less than 22.

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if a, b, c are the length of sides of a triangle, is

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