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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # The sides of a parallelogram are 10 and 11. The integral value of the

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Math Expert V
Joined: 02 Sep 2009
Posts: 65807
The sides of a parallelogram are 10 and 11. The integral value of the  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 63% (01:47) correct 37% (01:23) wrong based on 62 sessions

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The sides of a parallelogram are 10 and 11. The integral value of the length of the diagonal opposite to the acute angle is at most

A. 13
B. 14
C. 15
D. 19
E. 20

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Manager  G
Joined: 05 Jan 2020
Posts: 142
Re: The sides of a parallelogram are 10 and 11. The integral value of the  [#permalink]

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Maximum length of the diagonal will occur when the acute angle is close to 90

=> Maximum length of the diagonal will be the integer value less than the diagonal value if it were a right angle.

=> Maximum length = [root(10^2+11^2)] = [root(221)] = 14

[] - denotes the lower integral value.

Ans: B
PS Forum Moderator P
Joined: 18 Jan 2020
Posts: 1529
Location: India
GPA: 4
Re: The sides of a parallelogram are 10 and 11. The integral value of the  [#permalink]

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Let the triangle be a right angle triangle.
Therefore,
Side1^2 + Side2^2 = Diagonal^2
10^2 + 11^2 = Diagonal^2
100+121 = Diagonal^2
221 = Diagonal^2
Diagonal = 14.86 (approx)

Since diagonal is of the acute angle. It has to be least than 14.86
The maximum integral value of the diagonal less than 14.86 is 14

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Intern  B
Joined: 11 Feb 2017
Posts: 17
Re: The sides of a parallelogram are 10 and 11. The integral value of the  [#permalink]

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yashikaaggarwal wrote:
Let the triangle be a right angle triangle.
Therefore,
Side1^2 + Side2^2 = Diagonal^2
10^2 + 11^2 = Diagonal^2
100+121 = Diagonal^2
221 = Diagonal^2
Diagonal = 14.86 (approx)

Since diagonal is of the acute angle. It has to be least than 14.86
The maximum integral value of the diagonal less than 14.86 is 14

Posted from my mobile device

how to calculate 14.86 from root(221)
Intern  B
Joined: 15 Jun 2019
Posts: 6
Re: The sides of a parallelogram are 10 and 11. The integral value of the  [#permalink]

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1
i don't even understand what the question is asking...
can anyone draw a graph please? Thank you.
Intern  B
Joined: 15 Jun 2019
Posts: 6
The sides of a parallelogram are 10 and 11. The integral value of the  [#permalink]

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AabhishekGrover wrote:
yashikaaggarwal wrote:
Let the triangle be a right angle triangle.
Therefore,
Side1^2 + Side2^2 = Diagonal^2
10^2 + 11^2 = Diagonal^2
100+121 = Diagonal^2
221 = Diagonal^2
Diagonal = 14.86 (approx)

Since diagonal is of the acute angle. It has to be least than 14.86
The maximum integral value of the diagonal less than 14.86 is 14

Posted from my mobile device

how to calculate 14.86 from root(221)

well, i guess since 15^2= 225 and 14^2 = 196, \sqrt{221} must between 14 and 15.
Manager  G
Joined: 05 Jan 2020
Posts: 142
Re: The sides of a parallelogram are 10 and 11. The integral value of the  [#permalink]

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maggiesomi wrote:
i don't even understand what the question is asking...
can anyone draw a graph please? Thank you.

Refer attached figures.

In fig 1, angle ABC is obtuse and angle BCD is acute.

We are asked to find the length of diagonal opposite to the acute angle. => Length of BD.

Now we need to understand how are the diagonals related to the angle opposite to them.

Fig 2 is a rectangle. => BD^2 = BC^2 + CD^2

If the angle increases to more than 90 degrees, then the length of diagonal BD will increase. Thus, BD^2 will no longer be equal to BC^2 + CD^2.

BD^2 > BC^2 + CD^2. This is true of diagonal AC in figure 1. AC^2 > AB^2 + BC^2

If the angle decreases to less than 90 degrees, then the length of diagonal BD will decrease. Thus, BD^2 < BC^2 + CD^2 in figure 1.

We need to find the maximum possible integral value of BD when the angle is acute. This will happen when the angle is almost but less than 90. So, the length of BD will be slightly less than what it would be if the angle were a right angle.

BD^2 < BC^2 + CD^2
=> BD^2 < 11^2 + 10^2
=> BC^2 < 121 + 100
=> BC^2 < 221

We don't need to calculate the exact value of BD. We simply need to know the range of squares that it lies between.

=> 196 < BD^2 < 225
=> 14^2 < BD^2 < 15^2

=> The maximum integral value of BD (diagonal opposite the acute angle) will be 14.

Hope it helps!
Attachments Untitled.png [ 6.49 KiB | Viewed 310 times ]

Manager  B
Joined: 10 Oct 2019
Posts: 53
Location: India
Schools: NTU '21
GMAT 1: 530 Q40 V23
Re: The sides of a parallelogram are 10 and 11. The integral value of the  [#permalink]

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Bunuel wrote:
The sides of a parallelogram are 10 and 11. The integral value of the length of the diagonal opposite to the acute angle is at most

A. 13
B. 14
C. 15
D. 19
E. 20

for approximate value we can consider diagonals bisects each other at 90
so 10^2=8^2+6^2

so diagonal = 8+6=14 approx
ANS:B Re: The sides of a parallelogram are 10 and 11. The integral value of the   [#permalink] 10 Jul 2020, 20:43

# The sides of a parallelogram are 10 and 11. The integral value of the   