The area of the shaded region is 12 and the sum of a side of the outer

Let x be the side of outer square and y be the side of inner square
x^2-y^2=12
x+y=6
Solving above x=4;y=2
area of square= 16

Let's assume that side of outer square equals X and side of inner square equals Y
As per question,
X + Y = 6 ------(1)
\(X^2 - Y^2 = 12\)-----(2)

(X+Y) (X-Y) = 12
6.(X-Y) = 12
(X-Y) = 2

So X equals 4. Thus the area of outer square is 16.

Hence Answer D

Working from the answer choices.

Let X = outer square side and Y = inner square side

X + Y = 6

Area of shaded portion = 12. (Area of outer square - Area of inner square) = area of shaded portion ( = 12)

Find an area whose side will work with first equation, derive value of Y, then test both to see if derived values satisfy the second (area of shaded portion)

1. (A) and (B) are out because their areas, and hence sides, are too large. At a glance: \(\sqrt{36}\)= 6.*

Prompt says X + Y = 6. If X = 6, Y = 0. Not possible. So if the side X = 6 for area of 36 is too big, whatever side X is for area of 48 will be even bigger.

*Area of square is s\(^2\). So 36 = s\(^2\), \(\sqrt{36}\) = s, \(\sqrt{36}\) = 6; that's already too big, so \(\sqrt{48}\) is out, too.

2. (C), (D), and (E)? (D) is the only perfect square. Smart to start there; its square root, X, will be an integer.

If area of outer square is 16, 16 = s\(^2\), \(\sqrt{16}\) = s = 4.

A = 4
A + B = 6
B = 2

"[A]rea of the shaded region is 12."

Area of shaded region = (outer square area - inner square area).

Outer square area is 16. Inner square area is B\(^2\) = 2\(^2\) = 4

(Outer square area - inner square area) ==> (16 - 4) = 12, which = area of shaded region. Answer is

We can let the side of the inner square = y and the side of the outer square = x.

Since the sum of the sides of the two squares is 6, we have:

x + y = 6

Since the area of the shaded region is 12, we have:

x^2 - y^2 = 12

(x + y)(x - y) = 12

6(x - y) = 12

x - y = 2

Adding together x + y = 6 and x - y = 2, we have:

2x = 8

x = 4, so the area of the outer square is 4^2 = 16.

Answer: D