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Joined: 10 Apr 2015
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The two identical squares shown above are the largest square
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13 May 2020, 08:06
13 May 2020, 08:06
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The two identical squares shown above are the largest squares that can be drawn inside the semicircle. If the length of AB = √12 - √6, what is the area of rectangle BCDE?
A) 6
B) 2√6
C) 6√2
D) 12
E) 12√2
Re: The two identical squares shown above are the largest square
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13 May 2020, 12:06
13 May 2020, 12:06
1
Let the side of the square be a.
The diagonal of the square will be √2a.
AB+a=√2a
Substituting the value of AB will give "a" as √24- √6
Hence area of the rectangle will be 2*(√24- √6)^2=12
Option D
The diagonal of the square will be √2a.
AB+a=√2a
Substituting the value of AB will give "a" as √24- √6
Hence area of the rectangle will be 2*(√24- √6)^2=12
Option D
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: The two identical squares shown above are the largest square
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13 May 2020, 14:20
13 May 2020, 14:20
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GreenlightTestPrep wrote:
The two identical squares shown above are the largest squares that can be drawn inside the semicircle. If the length of AB = √12 - √6, what is the area of rectangle BCDE?
A) 6
B) 2√6
C) 6√2
D) 12
E) 12√2
Draw a diagonal in one of the squares
Let x = the length of each side of the square
Since we have a right triangle we can apply the Pythagorean Theorem to write: x² + x² = hypotenuse²
Simplify: 2x² = hypotenuse²
Solve: hypotenuse = (√2)x
IMPORTANT: This tells us that the radius of the circle has length (√2)x
Now that we have one way to express the circle's radius, notice that we have another way to find the circle's radius.
Since x = the length of one side of a square, and since AB = √12 - √6, we can see that the SUM of those two lengths = the circle's radius
So, we can write: The circle's radius = x + √12 - √6
Now that we have two different ways to express the circle's radius, we can write: (√2)x = x + √12 - √6
Subtract x from both sides of the equation to get: (√2)x - x = √12 - √6
Factor both sides: x(√2 - 1) = √6(√2 - 1)
Divide both sides by (√2 - 1) to get: x = √6
Since x is the length of one side of a square, the area of ONE square = (√6)(√6) = 6
So the area of BOTH squares = 6 + 6 = 12
Answer: D
Cheers,
Brent
