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What is the perimeter of a rectangle with integer sides, an area of 10

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What is the perimeter of a rectangle with integer sides, an area of 10  [#permalink]

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New post 29 Aug 2018, 03:54
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Re: What is the perimeter of a rectangle with integer sides, an area of 10  [#permalink]

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New post 29 Aug 2018, 04:45
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Diagonal = \(\sqrt{L^2+W^2}\) = 15

\(L^2\)+\(W^2\) = 225

Area = LW = 108

\((L+W)^2\) = \(L^2\)+\(W^2\)+2LW

\((L+W)^2\) = 225+216

L+W = 21

Perimeter = 2(L+W) = 2*21 = 42

D is the answer.
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Re: What is the perimeter of a rectangle with integer sides, an area of 10  [#permalink]

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New post 29 Aug 2018, 06:30
Bunuel wrote:
What is the perimeter of a rectangle with integer sides, an area of 108 square meters and a diagonal of 15 meters?


A. 9
B. 12
C. 18
D. 42
E. It cannot be determined from the information given.


Afc0892 has posted the correct solution but I will take a different approach to explore the question in another way.

The question states that the sides are integers. So, if we assume the sides to be 'a' & 'b' then we can see two things

First
a * b = 108, in which a & b can take different values such as
108 * 1 = 108
54 * 2 = 108
27 * 4 = 108
18 * 6 = 108
12 * 9 = 108

Second
A right-angled triangle with hypotenuse 15 and other sides a & b forming a right angle between them.
Now, in a triangle, a side has to be smaller than the sum of other two sides and greater than the positive difference of other two sides.
Mathematically speaking the Hypotenuse 15 should lie between
a + b > 15 > a - b

If we check from the above values, we can see that the property is satisfied only by the last two sets
That is two triangles can be made
( 18 , 6 , 15 ) and (12, 9 , 15)

Again, the hypotenuse has to be the longest side of the triangle so we can reject the first set (18, 6, 15) and land into our desired triangle (12, 9, 15)

So, sides are 12 and 9
Therefore perimeter is = 2 (12 + 9 ) = 42

this was a long story approach

The fun method is knowing Pythagorean triplets
The moment you see hypotenuse as 15 and it is given that the other two sides are integers then it must fall into the category of Pythagorean triplet satisfying the base version of (3, 4, 5)

Further examples are (5, 12, 13), (8, 15, 17), (7, 24, 25)
Given that hypotenuse is 26 and other two sides are integers then they must fall into the category (5, 12, 13)

Now, you should also convince yourself that the two sides if not integers can also satisfy Pythagoras theorem. What I mean is that if it is not mentioned that the other two sides are integers then do not fit triplet strategy un-necessarily.

Because,
12^2 + 9^2 = 15^2

Similarly,
10.198^2 + 11^2 = 15^2

It is better to visualize. I found this link
https://www.calculator.net/triangle-cal ... &x=76&y=25
Hit and see different combinations of a triangle.

I know few of you may think it is all gibberish and why am I making it complex if it can be solved easily :)
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What is the perimeter of a rectangle with integer sides, an area of 10  [#permalink]

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New post 29 Aug 2018, 06:32
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Bunuel wrote:
What is the perimeter of a rectangle with integer sides, an area of 108 square meters and a diagonal of 15 meters?


A. 9
B. 12
C. 18
D. 42
E. It cannot be determined from the information given.


Let x = the length of the rectangle's base
Let y = the height of the rectangle

GIVEN: area = 108
area = (base)(height)
So, 108 = xy

GIVEN: diagonal = 15
The diagonal creates a RIGHT TRIANGLE with legs x and y and hypotenuse 15
So, we can apply the Pythagorean Theorem to get: x² + y² = 15²
Evaluate to get: x² + y² = 225

What is the perimeter of the rectangle?
In other words, we want to find the value of 2(x + y)

Here's how we can do this quickly:
First notice that we already have most of what it takes to create the special product x² + 2xy + y²
Why is this useful?
Well, we know that x² + 2xy + y² = (x + y)², and it would be very useful to determine the value of x + y to help us determine the perimeter.

So, here's what we'll do:
If 108 = xy, then 216 = 2xy, which we'll rewrite as: 2xy = 216

We have:
2xy = 216
x² + y² = 225

Add the equations to get: x² + 2xy + y² = 441
Factor the left side: (x + y)² = 21²
This means x + y = 21
And this means 2(x + y) = 42

In other words, the perimeter is 42

Answer: D

Cheers,
Brent
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What is the perimeter of a rectangle with integer sides, an area of 10  [#permalink]

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New post 29 Aug 2018, 14:38
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Bunuel wrote:
What is the perimeter of a rectangle with integer sides, an area of 108 square meters and a diagonal of 15 meters?


A. 9
B. 12
C. 18
D. 42
E. It cannot be determined from the information given.


Diagonal of a rectangle is the hypotenuse of a right angled triangle whose sides are integers.hence its sides are in the form 3x:4x:5x (This came into mind when I saw 15 and it's a multiple of 5)

hypotenuse=5x=15 or, x=3

So, sides of rectangle are : 9 and 12 (Area=12*9=108 matches with the given info)

So, perimeter=2(9+12)=42

Ans. (D)
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Re: What is the perimeter of a rectangle with integer sides, an area of 10  [#permalink]

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New post 03 Sep 2018, 18:57
Bunuel wrote:
What is the perimeter of a rectangle with integer sides, an area of 108 square meters and a diagonal of 15 meters?


A. 9
B. 12
C. 18
D. 42
E. It cannot be determined from the information given.


We can create the equation for the area as:

Area = LW = 108

We use the Pythagorean theorem to create the equation for the length of the diagonal.

L^2 + W^2 = 15^2

Recall that (L + W)^2 = L^2 + W^2 + 2WL, so we have:

(L + W)^2 = 15^2 + 2(108)

(L + W)^2 = 225 + 216

(L + W)^2 = 441

Taking the square root of both sides, we have:

L + W = 21

Recall that the perimeter of a rectangle is P = 2(L + W), so we have P = 2(21) = 42.

Answer: D
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Re: What is the perimeter of a rectangle with integer sides, an area of 10   [#permalink] 03 Sep 2018, 18:57
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