A painting at an art gallery is framed such that the area

Question

A painting at an art gallery is framed such that the area of the square frame is 1/4 the area of the painting itself. If the diagonal line connecting corners of the frame has a length of 10, what is the area of the painting inside the frame?

A. 10
B. 20
C. 30
D. 40
E. 50

A. 10
B. 20
C. 30
D. 40
E. 50

BrushMyQuant · Answer

Given that A painting at an art gallery is framed such that the area of the square frame is 1/4 the area of the painting itself. And we need to find If the diagonal line connecting corners of the frame has a length of 10, what is the area of the painting inside the frame?

Figure attached

Suppose side of the painting(which is a square) is "a" and side of the outer square(painting + frame) is "b"

Area of painting  = a^2 and we know that the area of the frame is (1/4) of that of the painting so
 Area of frame  = (1/4) * a^2

Area of Frame + Painting  = a^2 + (1/4)*a^2 = (5/4)*a^2 which is equal to b^2

Line connecting the corners of the frame is the diagonal of the outer square which is equal to b*sqrt2

so, b * sqrt2 = 10
so, b = 5 * sqrt2
b^2 = 50

we know that b^2 = (5/4)*a^2
so, a^2 = (4/5)*b^2 = (4/5)*50 = 40

And area of painting = a^2 = 40

So,  Answer will be D .
Hope it helps!

nave · Answer

It has been established that both the painting and the frame are square.

So let the side of the frame be 'A' and the side of the painting be 'a'

It is stated that  A^2-a^2 = \frac{1}{4}*a^2

It has also stated that the diagonal of the frame is 10, so  \sqrt{2}*A = 10

So  A=\frac{10}{\sqrt{2}}  which implies  A^2=50

50-a^2 = \frac{1}{4}*a^2  
 50 = \frac{5}{4}*a^2 
So,  a^2 = 40

Answer is D

KarishmaB · Answer

Start the solution with what is given to you in the question. You may not need to use any variables in that case.

Diagonal = 10
Then, side of the frame =  5\sqrt{2} 
Area of the entire painting + frame =  (5\sqrt{2})^2 = 50

Since area of frame is 1/4th of the area of painting, 
Area of frame = 10 and Area of painting = 40

gmacforjyoab · Answer

Oh , that's such a neat way !

holidayhero · Answer

Let d=diagonal length
Area of a square =  \frac{d^2}{2}

So the entire area of the Frame + Painting would be. 
 A= \frac{10^2}{2} 
 A= \frac{100}{2} 
 A= 50

We know that the area of the frame is 1/4 that of the painting. So it is clear that the frame is 10 and the painting is 40.

dcummins · Answer

Apologies I understand it a bit better now, but i was confused by the question wording.

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A painting at an art gallery is framed such that the area

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