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# The three sides of an acute angled triangle-12,19&X. How many X?

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The three sides of an acute angled triangle-12,19&X. How many X?  [#permalink]

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26 Jun 2016, 06:07
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The three sides of an acute angled triangle (all in cms) are 12, 19 and x. How many integer values can x take?

A. 23
B. 22
C. 15
D. 8
E. 7

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The three sides of an acute angled triangle-12,19&X. How many X?  [#permalink]

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26 Jun 2016, 06:38
2
In acute angle triangles all angles in the triangle should be less than 90.

The maximum length of x can be when it is the hypotenuse of the triangle.
So, x= sqrt(12^2 + 19^2) = 22.5 (approx)

The maximum length of x can be when side of length 12 is the hypotenuse of the triangle.
So, x = sqrt (19^2 - 12^2) = 14.7 (approx)

So, x>14.7 and x<22.5.

Hence, possible values for x can be 15,16,17,18,19,20,21 and 22. Total 8 values.

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Re: The three sides of an acute angled triangle-12,19&X. How many X?  [#permalink]

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26 Jun 2016, 07:15
Hi, Thank you for the prompt response!!

Should the Third paragraph read - The Minimum length of X can be when 19 is the Hypotenuse of the Triangle?

Do correct me if I am wrong here.

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Re: The three sides of an acute angled triangle-12,19&X. How many X?  [#permalink]

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26 Jun 2016, 08:27
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Re: The three sides of an acute angled triangle-12,19&X. How many X?  [#permalink]

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26 Jun 2016, 10:59
For all triangle any side is greater than the difference of other two and less that the sum
so 12+19 = 31
12-19=7
so 7<x<31=(31-7)+1
23

A
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Re: The three sides of an acute angled triangle-12,19&X. How many X?  [#permalink]

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26 Jun 2016, 11:12
Hi LogicGuru1,

Pls refer to the links shared by Bunuel above. It has in-depth discussion on this topic.

I am sure, you will find it useful!

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Re: The three sides of an acute angled triangle-12,19&X. How many X?  [#permalink]

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26 Jun 2016, 11:15
Hi Abhishek,

The catch here is "Acute triangle". So we need to do two steps of calculations.

Pls refer to above responses by Bunuel and 14101992.

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Re: The three sides of an acute angled triangle-12,19&X. How many X?  [#permalink]

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26 Jun 2016, 11:15
14101992 wrote:
In acute angle triangles all angles in the triangle should be less than 90.

The maximum length of x can be when it is the hypotenuse of the triangle.
So, x= sqrt(12^2 + 19^2) = 22.5 (approx)

The maximum length of x can be when side of length 12 is the hypotenuse of the triangle.
So, x = sqrt (19^2 - 12^2) = 14.7 (approx)

So, x>14.7 and x<22.5.

Hence, possible values for x can be 15,16,17,18,19,20,21 and 22. Total 8 values.

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By definition an acute triangle is one that has all three internal angles less than 90 degrees
[/u]
Aren't you using Pythagorus theorem which is only and exclusively meant for right triangle. (having one intnernal angle that measure exactly 90 degrees )
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Re: The three sides of an acute angled triangle-12,19&X. How many X?  [#permalink]

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26 Jun 2016, 11:52
14101992 wrote:
In acute angle triangles all angles in the triangle should be less than 90.

The maximum length of x can be when it is the hypotenuse of the triangle.
So, x= sqrt(12^2 + 19^2) = 22.5 (approx)

The maximum length of x can be when side of length 12 is the hypotenuse of the triangle.
So, x = sqrt (19^2 - 12^2) = 14.7 (approx)

So, x>14.7 and x<22.5.

Hence, possible values for x can be 15,16,17,18,19,20,21 and 22. Total 8 values.

-------------------------------------

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You mean one maximum and one minimum right?

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Re: The three sides of an acute angled triangle-12,19&X. How many X?  [#permalink]

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26 Jun 2016, 21:11
LogicGuru1

You are right. Pythagoras theorem is used for only right angle triangles and a right angle triangle has one angle as 90 and rest two <90.

Now, you are given to find out the all possible acute angle triangles. Acute angle triangle will have all angles less than 90. So, by using Pythagoras theorem, we will have hypotenuse of that length where an angle of the triangle will be 90 and rest 2 angles less than 90.

Now, as you can see in my previous post, x>14.7 and x<22.5 and not x≥14.7 and x≤22.5. If I would have taken the latter, then I would be including the triangles with one angle as 90 degrees. But, since I have taken the former, it will only give me values of x for which all angles will be less than 90.

I hope I am able to solve your query.

paidlukkha
The option A will include many triangle which have one of the angles that are greater than 90. Kindly go through the solution one more time. You will be able to get it.

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Re: The three sides of an acute angled triangle-12,19&X. How many X?  [#permalink]

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19 Jul 2017, 08:33
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Re: The three sides of an acute angled triangle-12,19&X. How many X?   [#permalink] 19 Jul 2017, 08:33
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