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555-605 Level|   Geometry|                           
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jabhatta2
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What is the area of rectangular region R? 21 Apr 2020, 06:06
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Bunuel
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Bunuel,

isn't this a special Pythagorean triangle? with a,b,c 3,4,5? Hence it would be A not C...

This is a common trap.

Knowing that the hypotenuse equals to 5 DOES NOT mean that the sides of the right triangle necessarily must be in the ratio of Pythagorean triple - 3:4:5. Or in other words: if \(x^2+y^2=5^2\) DOES NOT mean that \(x=3\) and \(y=4\). Certainly this is one of the possibilities but definitely not the only one. In fact \(x^2+y^2=5^2\) has infinitely many solutions for \(x\) and \(y\) and only one of them is \(x=3\) and \(y=5\).

For example: \(x=1\) and \(y=\sqrt{24}\) or \(x=2\) and \(y=\sqrt{21}\)...

Hope this helps.
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Hi Bunuel - i thought the answer might be A because

-- the diagonal breaks up the 90 degrees into 2 equal halves (45 degrees each)

hence i thought the diagonal breaks up the rectangle into 2 triangles that are 45-45-90 triangles

hence, i thought this was a special triangle with the ratio of the sides : 1:1 : square root 2

hence you can find the length and breadth

given area is length * breadth = you can then find the area of the rectangle

hence i marked (A) as sufficient

Could you let me know where is the break in my logic
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What is the area of rectangular region R? 21 Apr 2020, 07:06
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jabhatta@umail.iu.edu
Bunuel
unceldolan
Bunuel,

isn't this a special Pythagorean triangle? with a,b,c 3,4,5? Hence it would be A not C...

This is a common trap.

Knowing that the hypotenuse equals to 5 DOES NOT mean that the sides of the right triangle necessarily must be in the ratio of Pythagorean triple - 3:4:5. Or in other words: if \(x^2+y^2=5^2\) DOES NOT mean that \(x=3\) and \(y=4\). Certainly this is one of the possibilities but definitely not the only one. In fact \(x^2+y^2=5^2\) has infinitely many solutions for \(x\) and \(y\) and only one of them is \(x=3\) and \(y=5\).

For example: \(x=1\) and \(y=\sqrt{24}\) or \(x=2\) and \(y=\sqrt{21}\)...

Hope this helps.

Hi Bunuel - i thought the answer might be A because

-- the diagonal breaks up the 90 degrees into 2 equal halves (45 degrees each)

hence i thought the diagonal breaks up the rectangle into 2 triangles that are 45-45-90 triangles

hence, i thought this was a special triangle with the ratio of the sides : 1:1 : square root 2

hence you can find the length and breadth

given area is length * breadth = you can then find the area of the rectangle

hence i marked (A) as sufficient

Could you let me know where is the break in my logic
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The diagonal of e rectangle bisects the angle ONLY IF the rectangle is a square. If a rectangle is NOT a square, then the diagonal can create an angle with an adjacent side, from 0 to 90, not inclusive.
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What is the area of rectangular region R? 03 Jun 2020, 04:40
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Bunuel
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I had read a formula somewhere which said if any quadrilateral has diagonals intersecting at 90 degrees, then their area can be found out by 1/2 * d1 * d2.
If we use this formula, then A alone is sufficient to answer the question.
Where am I going wrong Bunuel ? Is that formula right?

Do we know that the diagonals intersect at 90 degrees from (1)?

You're right, "rectangular" doesn't necessarily mean a rectangle. Missed that.

No, that's not the point. Rectangular region does mean that the figure is a rectangle. But the diagonals of a rectangle do not always intersect at 90 degrees. This happnes only when a rectangle is a square.
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If the two diagonal are the same does not mean the rectangular is a square?
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What is the area of rectangular region R? 03 Jun 2020, 04:57
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Eduardo86
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Rajeet123
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Rajeet123
I had read a formula somewhere which said if any quadrilateral has diagonals intersecting at 90 degrees, then their area can be found out by 1/2 * d1 * d2.
If we use this formula, then A alone is sufficient to answer the question.
Where am I going wrong Bunuel ? Is that formula right?

Do we know that the diagonals intersect at 90 degrees from (1)?

You're right, "rectangular" doesn't necessarily mean a rectangle. Missed that.

No, that's not the point. Rectangular region does mean that the figure is a rectangle. But the diagonals of a rectangle do not always intersect at 90 degrees. This happnes only when a rectangle is a square.

If the two diagonal are the same does not mean the rectangular is a square?
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The diagonals of any rectangle are equal. So, not, equal diagonal does not necessarily mean that the rectangle is a square.
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What is the area of rectangular region R? 05 Jul 2021, 05:58
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Bunuel
What is the area of rectangular region R?

(1) Each diagonal of R has length 5.
(2) The perimeter of R is 14.
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Solution:

Question Stem Analysis:

We need to determine the area of region R, which is a rectangle.

Statement One Alone:

Knowing the length of the diagonal of a rectangle is not sufficient to determine its area. For example, if R is a square (recall that a square is also a rectangle), it’s side length would be 5/√2 and its area would be (5/√2)^2 = 25/4. However, R could also be a non-square rectangle with dimensions 3 and 4. In this case, the area of R is 3 x 4 = 12. Statement one alone is not sufficient.

Statement Two Alone:

Knowing the perimeter of a rectangle is not sufficient to determine its area. For example, if R has dimensions 2 and 5, its area is 2 x 5 = 10. However, if R has dimensions 3 and 4, its area is 3 x 4 = 12. Statement two alone is not sufficient.

Statements One and Two Together:

If we let L and W be the dimensions of the rectangle, we can create the equations:

L^2 + W^2 = 5^2

and

2L + 2W = 14

We see that we can solve L and W based on the equations we’ve set up, and once we have the values of L and W, then the area of region R is just the product of L and W. Both statements together are sufficient. (Note: L = 3 and W = 4 OR L = 4 and W = 3. Either way, the area of R is 12.)

Answer: C
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What is the area of rectangular region R? 01 Aug 2025, 01:43
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