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29 Aug 2018, 03:54
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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.
A. 9
B. 12
C. 18
D. 42
E. It cannot be determined from the information given.
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.
\(L^2\)+\(W^2\) = 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.
irishiraj87
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29 Aug 2018, 06:30
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Bunuel
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.
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
