If we are able to get a UNIQUE value for the area of the triangle from the information given in the statements, the data is sufficient.
Check to see whether more than one set of values are possible with the given data. For e.g., more than one right triangle can be described for a given hypotenuse of 10 units.
Remember that sides of a triangle need not necessarily be integers. Particularly, one might be tempted to think that only triangles whose sides are Pythagorean triplets are right triangles. All Pythagorean triplets are right triangles. But all right triangles need not be Pythagorean triplets. For instance 1, 1, \\sqrt 2 \\) will form sides of a right triangle, where one of the sides is not an integer.
It is not possible to find the area of a right triangle with information about just one of its sides and no other information about the triangle.
Statement 1 alone is NOT sufficient to answer the question.
Because statement 1 is NOT sufficient, we can eliminate choices A and D.
Answer choices narrow down to B, C or E.
The temptation is to conclude that the sides are 3, 4, and 5. These values will form sides of a right triangle and the sum of a, b, and c is 12.
But before concluding that we have a definite answer, let us check whether any right triangle other than 3, 4, and 5 is possible.
For instance, it is possible to have a right isosceles triangle whose perimeter is 12. The area of that triangle will be different from that of the triangle with sides 3, 4, and 5.
We are not getting a UNIQUE value for the area of the triangle from the information in statement 2. Statement 2 alone is NOT sufficient to answer the question.
Because statement 2 is NOT sufficient, we can eliminate choice B as well.
Answer choices narrow down to C or E.
In a right triangle, the longest side is the hypotenuse. Side that measures 4 is therefore, not the hypotenuse as 4 is the average of the sum of the 3 sides.
The hypotenuse has to necessarily be greater than the average of the 3 sides. So, either b or c is the hypotenuse.
Let us assume c to be the hypotenuse. So, 4 and b are the two perpendicular sides of the right triangle.
4 + b + c = 12. So, b + c = 8. Therefore, b = 8 - c
Applying Pythagoras theorem, c^{2} = 4^{2} + (8 - c)^{2}
Solving for c, c^{2} = 16 + 64 - 16c + c^{2}
16c = 80. So, c = 5.
If c = 5, b = 8 - c = 3
Using the two statements, we could a unique set of values for a, b, and c. Hence, we will be able to find the area of the triangle.
The two statements TOGETHER are sufficient to answer the question.
Choice C is the correct answer.