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655-705 (Hard)|   Geometry|         
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12 Days of Christmas GMAT Competition - Day 10: Is the perimeter of a 23 Dec 2020, 19:00
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12 Days of Christmas GMAT Competition with Lots of Fun

Is the perimeter of a rectangle more than 60 cm ?

(1) The area of the rectangle is 226 cm^2.
(2) The length of a diagonal of the rectangle is 30 cm.




M36-92

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Bunuel
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12 Days of Christmas GMAT Competition - Day 10: Is the perimeter of a 12 Nov 2021, 07:43
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Bunuel
12 Days of Christmas GMAT Competition with Lots of Fun

Is the perimeter of a rectangle more than 60 cm ?

(1) The area of the rectangle is 226 cm^2.
(2) The length of a diagonal of the rectangle is 30 cm.




M36-92
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Is the perimeter of a rectangle more than 60 cm ?

(1) The area of the rectangle is 226 cm^2.

Given \(ab = 226\).

Now, for rectangles with a fixed area, a square has the minimum perimeter. So, the minimum perimeter will occur when \(a=b\). In this case, we'd have \(a^2=226\).

\(a=15.something\) and thus the perimeter is \(4*15.something=60.something>60\). As even the minimum possible perimeter is more than 60 cm, then the actual perimeter, whatever it is, is also more than 60 cm. Sufficient.

(2) The length of a diagonal of the rectangle is 30 cm.

The diagonal, the length and the width of a rectangle form a triangle and in a triangle the length of any side must be smaller than the sum of the other two sides. So, \(a+b>30\) and thus \(perimeter=2(a + b) > 60\) Sufficient.


Answer: D
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12 Days of Christmas GMAT Competition - Day 10: Is the perimeter of a Updated on: 26 Dec 2020, 00:47
Originally posted by KumarSri on 23 Dec 2020, 19:24.
Last edited by KumarSri on 26 Dec 2020, 00:47, edited 1 time in total.
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statement 1:

Area is 226 cm^2

assume Square, l=b
then l^2 = 226
l=15.xx (approx)

so perimeter 4l is more than 60. For rectangle then l is greater than breadth. so obviously perimeter is greater than 60

Hence Statement 1 is sufficient

Statement 2:

hypotenuse is 30. That means sum of two sides should be greater than 30. Hence perimeter will be greater than 60.

So Statement 2 is also sufficient.


Hence option D
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12 Days of Christmas GMAT Competition - Day 10: Is the perimeter of a 23 Dec 2020, 19:40
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My answer is (D): Either (1）or (2) is SUFFICIENT.

(1) The area of the rectangle is 226 cm^2.
Consider a square with 60 cm perimeter. Its area is 15 * 15 = 225, which is smaller than 226.
And we know that for a given rectangle of 60 cm, square yields the largest area.
So, if the area of the rectangle is 226 cm^2, its rectangle must be more than 60cm. (Otherwise, the maximum area is 225 cm^2)
SUFFICIENT

(2) The length of a diagonal of the rectangle is 30 cm.
That means the sum of the length and height of the rectangle is > 30cm.
So the perimeter of a rectangle is > 60cm.
SUFFICIENT.
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12 Days of Christmas GMAT Competition - Day 10: Is the perimeter of a 23 Dec 2020, 20:06
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Is the perimeter of a rectangle more than 60 cm ?

(1) The area of the rectangle is 226 cm^2.
(2) The length of a diagonal of the rectangle is 30 cm.

L + W > 30 Y/N?


1) Area = 226 = LxW

We have (L+W)^2 >= 4LW = 4x226 > 4x225 = (2x15)^2 --> L + W > 30 --> Yes --> Sufficient
Notice that: (L+W)^2 >= 4LW or (L-W)^2 >= 0 is always true


2) Diagonal = 30

Let's consider half of that rectangle, we will have L + W > Diagonal = 30 (according to side inequalities in a certain triangle)
--> Yes --> Sufficient

Correct answer: D
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12 Days of Christmas GMAT Competition - Day 10: Is the perimeter of a 23 Dec 2020, 22:50
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Is the perimeter of a rectangle more than 60 cm ?

(1) The area of the rectangle is 226 cm^2.

Concept: For a given area of quadrilateral, a square would have the smallest perimeter.
So, let us assume that the rectangle is actually a square.
If area (of square) = 226, side = 15.033 (approx)
So the perimeter would be slightly greater than 60 cm.
If a square has a greater perimeter than 60, a rectangle would definitely have a perimeter > 60
Sufficient.


(2) The length of a diagonal of the rectangle is 30 cm.

If length = l and Breadth = b
\(l^2 + b^2\) = 900
Suppose: b= 20, then l= 22.36. (perimeter 84.72 > 60)
Suppose: b= 10, then l= 28.28. (perimeter 76.56 > 60)
Suppose: b= 1, then l= 29.98. (perimeter 61.96 > 60)
Now Suppose b = 0.0001, then l = 29.9999983333 (perimeter 60.000196 > 60)
As one of the sides gets smaller and smaller, the perimeter gets closer to 60.
HOWEVER, the perimeter will ALWAYS be > 60,
This is also sufficient

Both statements individually sufficient.
Answer D
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12 Days of Christmas GMAT Competition - Day 10: Is the perimeter of a 04 Sep 2023, 02:40
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