What is the area of the rectangular region depicted above with sides

We require \(A\) and \(B\) to determine the area of the rectangle.

\(\textbf{(1) } C^2 = 9 + A^2\)

By Pythagoras: \(C^2 = A^2 + B^2\)

\(B^2 = 9\\\\
B = 3\)

We do not know \(A\).

Insufficient

\(\textbf{(2) } A + B = 7\)

\(A\) and \(B\) could be any values that add up to 7, we can determine the perimeter, but not the area.

Insufficient

\(\textbf{(1 + 2)}\)

\(B = 3 \\\\
A + B = 7\\\\
A = 4\)

Sufficient As we have \(A\) and \(B\).

shouldn't the answer be A.
As C^2-A^2=9
So (C-A)(C+A)=9
which has only two values which satisfy C=5 and A=4

Answer A

The variables are not restricted to integers. \(C = 10, A = \sqrt{91}\) would satisfy the equation.

What is the area of the rectangular region depicted above with sides A and B and diagonal C?

(1) C^2 = 9 + A^2

(2) A + B = 7

A) Since C is the hyp : C^2 = 3^2 + A^2
(3,4,5) is a the pyth triplet. We can consider it : in that case: A = 4 and and area is 12
But it is not given that the lengths of the sides are int's ------> hence we could have some fractions for A and C satisfying C^2 = 9 + A^2 -----> insufficient

B) A + B = 7 ------> Could be (A, B) ----> (1 , 7) : Area = 7
(2, 5) : Area = 10 or (3, 4) Area : 12

C) B = 3 and A + B = 7 hence A = 4 hence area is 12 -----> suff

What is the area of the rectangular region depicted above with sides A and B and diagonal C?

(1) C^2 = 9 + A^2

(2) A + B = 7


In order to solve this question we need the width and length of the rectangle; in other words, we need to know the value of A and B.

Statement (1) gives us two variables so we cannot solve for either C or A. There is also no way to calculate B from this statement.

Statement (2) tells us that A + B = 7 - we can assume that B corresponds to the opposite side in the rectangle and use the Pythagorean theorem to calculate the hypotenuse

A^2 + B^2 =49 therefore C equals 7. Though we still have two variables, and no angles or angle ratios in the triangle and therefore cannot calculate either side A or B.

Statement (1) and (2) satisfy each other's missing variable. Therefore, Statement (1) and Statement (2) are sufficient.

area = AB
and C^2 =A^2 + B^2

St 1
C^2 = 9 +A^2
clearly we can see that B^2 = 9 or B=3. no idea about A. INSUFFICIENT

St 2: A+B=7. two variable, one equation. INSUFFICIENT

St 1 & St 2: A+B=7 and B=3. therefore A=4. hence AB = 12. ANSWER

Option C

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