In the figure above, if the area of triangular region D is 4

In the figure above, if the area of triangular region D is 4, what is the length of a side of square region A ?

The area of triangular region D = \(\frac{xy}{2} = 4\) --> \(xy=8\).
Question: \(z = \sqrt{x^2+y^2}\)

(1) The area of square region B is 9 --> \(x^2=9\) --> \(x=3\) --> \(y=\frac{8}{x}=\frac{8}{3}\). Sufficient.

(2) The area of square region C is 64/9 --> \(y^2=\frac{64}{9}\) --> \(y=\frac{8}{3}\) --> \(x=\frac{8}{y}=3\). Sufficient.

Answer: D.
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We have the area of right triangle D is 4, so if we know one of two legs, we could calculate the hypotenuse or the side of square region A ( using Pythagorean theorem c^2 = a^2 + b^2)

(1) From area of B, we could calculate one of the two legs -> sufficient
(2) From area of C, we could calculate one of the two legs -> sufficient

D is the answer

Hi Bunnel,
I wonder why aren't we using the hypotenuse as one of the sides when calculating the area??

Check here: math-triangles-87197.html
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Hi All,

Since this is a DS question, we CANNOT trust the picture. We can trust any descriptions and numbers that we are given though - so we know that the triangle is a RIGHT triangle, it's area IS 4 and the shape A IS a square. We're asked for a side length of square A.

1) The area of square region B is 9.

Since shape B is a square - and its area is 9 - we know that its sides are each 3. Knowing the height of the triangle is 3, we can figure out its base:

Area = (1/2)(Base)(Height)
4 = (1/2)(B)(3)
4 = 1.5B
4/1.5 = B

With the base and the height, we CAN figure out the length of the diagonal (by using the Pythagorean Theorem), so we would have the answer to the question.
Fact 1 is SUFFICIENT

2) The area of square region C is 64/9.

While the information in Fact 2 might look a big 'weird', we know from the work that we did in Fact 1 that if we have either the base or the height of the triangle, then we can figure out the other side and then solve the diagonal. Thus, Fact 2 is also enough information to answer the question.
Fact 2 is SUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich

why are dividing 8 by 3 can you explain the logic ? \(y=\frac{8}{x}=\frac{8}{3}\).

Hello

Bunuel explained in his solution that the area of the right triangle will be 1/2 * x * y or xy/2, this area is give as 4, so xy/2 = 4 or xy = 8 which in turn tells us that y = 8/x. He has explained in the highlighted part.

We have the area of right triangle D is 4, so if we know one of two legs, we could calculate the hypotenuse or the side of square region A ( using Pythagorean theorem c^2 = a^2 + b^2)
Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

dear Bunuel, I picked up E because I though there is no any information tells us B and C are square.
could you help clarify further?

thanks in advance

Check the highlighted parts. B is "square region" means that B is a square.
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Solution:

We need to determine the length of a side of square region A, given that the area of triangular region D is 4. Notice that region D is a right triangle. Therefore, if we know the lengths of the two legs of the triangle D, then by the Pythagorean theorem, we can determine the length of its hypotenuse, which is also a side of square A.

Statement One Alone:

Since the area of square region B is 9, its side length is 3. Since the area of triangle region D is 4, we can let x be the length of the side that is common to both regions D and C and create the equation (notice that the side is the base of the triangle):

½ * x * 3 = 4

3/2 * x = 4

x = 4 * ⅔ = 8/3

Recall that we determined that if we know the lengths of the two legs of triangle D, we can determine the length of a side of square A. Since now we have the lengths of both legs of triangle D, the length of a side of square A can be determined. Statement one alone is sufficient.

Statement Two Alone:

Since the area of square region C is 64/9, its side length is 8/3. Since the area of triangle region D is 4, we can let y be the length of the side that is common to both regions D and B and create the equation (notice that the side is the height of the triangle):

½ * 8/3 * y = 4

4/3 * y = 4

y = 4 * ¾ = 3

Recall that we determined that if we know the lengths of the two legs of triangle D, we can determine the length of a side of square A. Since we now know the lengths of both legs of triangle D, the length of a side of square A can be determined. Statement two alone is sufficient.

Answer: D

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The question doesn't really state if figure B and C are square or not, which we only get to know AFTER taking both options into account. So shouldn't the answer be C instead of D?

Hi chiragjn,

While we cannot trust that the drawing is necessarily drawn to scale, we CAN trust whatever facts the prompt gives us. Based on the prompt, we know 3 things for sure:

1) The triangle is a RIGHT TRIANGLE.
2) Based on the specific question that is asked, we know that region A is a SQUARE.
3) The AREA of the right triangle = 4... meaning (1/2)(Base)(Height) = 4

By themselves, both Fact 1 and Fact 2 gives us enough information to find the hypotenuse. Fact 1 can be used to find the Height of the triangle, which we can then use to find the Base (by using the area formula) and then the Hypotenuse (using the Pythagorean Theorem). Fact 2 can be used to find the Base of the triangle, which we can then use to find the Height (again, by using the area formula) and then the Hypotenuse (again, by using the Pythagorean Theorem). Thus, each Fact is sufficient on its own.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Video solution from Quant Reasoning:
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