Results 1 to 10 of 10

Thread: Triangle Inequality

  1. #1
    Banned
    Joined
    Oct 2012
    From
    Istanbul
    Posts
    680
    Thanks
    4

    Triangle Inequality

    ABC is a triangle. |BC|=a units,|CA|=b units, |AC|=c units.
    2a+b+c=24
    2a+2c=29
    How many integer values a can assume ?

    My work:
    a+c=14.5
    c-b=5
    a>c-b=5
    b+c>a
    b+c=24-2a
    24>3a
    8>a>5 is my answer and there are 2 values. But according to the book there are 3 values.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    22,429
    Thanks
    3314
    Awards
    1

    Re: Triangle Inequality

    Quote Originally Posted by kastamonu View Post
    ABC is a triangle. |BC|=a units,|CA|=b units, |AC|=c units.
    2a+b+c=24
    2a+2c=29
    How many integer values a can assume ?

    My work:
    a+c=14.5
    c-b=5
    a>c-b=5
    b+c>a
    b+c=24-2a
    24>3a
    8>a>5 is my answer and there are 2 values. But according to the book there are 3 values.
    If you can show that $a=5\text{ or }a=8$ both give a contradiction, then your answer is correct.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Oct 2012
    From
    Istanbul
    Posts
    680
    Thanks
    4

    Re: Triangle Inequality

    If I can show that a=8 or a=5 then there will be 3 values but I couldn't find a way.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    1,073
    Thanks
    549

    Re: Triangle Inequality

    Only two integers satisfy the condition $\displaystyle 5<a<8$

    so the problem has only two solutions $\displaystyle a=6$ and $\displaystyle a=7$
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2010
    Posts
    3,728
    Thanks
    1524

    Re: Triangle Inequality

    If $a=5$ then $2a+2c=10+2c=29$ implies $c=9.5$. Then $2a+b+c = 10+b+9.5=24$ implies $b=4.5$. But then $5+4.5 = a+b = c = 9.5$ is a contradiction to $ABC$ being a triangle.
    So, let's try $a=8$: Then $2a+2c = 16+2c = 29$ implies $c=6.5$. Then $2a+b+c = 16+b+6.5 = 24$ implies $b=1.5$. But then $1.5 + 6.5 = b+c = a = 8$ is a contradiction to $ABC$ being a triangle. Therefore, the only possible integer values are $a=6$ or $a=7$.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Banned
    Joined
    Oct 2012
    From
    Istanbul
    Posts
    680
    Thanks
    4

    Re: Triangle Inequality

    Then my answer is correct.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Aug 2006
    Posts
    22,429
    Thanks
    3314
    Awards
    1

    Re: Triangle Inequality

    Quote Originally Posted by kastamonu View Post
    If I can show that a=8 or a=5 then there will be 3 values but I couldn't find a way.
    I wrote "If you can show that $a=5\text{ or }a=8$ both give a contradiction, then your answer is correct."
    Did you read it?

    1) let $a=8$ you will get as consequence $b+c=a$ which is a contradiction.
    2) next let $a=5$ you will get as consequence $b+c=a$ which is a contradiction.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Banned
    Joined
    Oct 2012
    From
    Istanbul
    Posts
    680
    Thanks
    4

    Re: Triangle Inequality

    Yes I read it but it seems as a contradiction according to SlipEternal's post. So my answer seems to be correct.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor
    Joined
    Nov 2010
    Posts
    3,728
    Thanks
    1524

    Re: Triangle Inequality

    Quote Originally Posted by kastamonu View Post
    Yes I read it but it seems as a contradiction according to SlipEternal's post. So my answer seems to be correct.
    What Plato means is that he agreed with you from the beginning. He then gave you the method to use to show that your answer is correct. I then demonstrated that Plato's method to show that you are correct does, in fact, show that you are correct.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Joined
    Feb 2015
    From
    Ottawa Ontario
    Posts
    2,255
    Thanks
    500

    Re: Triangle Inequality

    2 solutions ONLY:
    a-b-c
    6-3.5-8.5
    7-2.5-7.5
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Oct 26th 2013, 02:10 AM
  2. Triangle inequality
    Posted in the Geometry Forum
    Replies: 8
    Last Post: Mar 6th 2011, 06:40 AM
  3. Replies: 3
    Last Post: Dec 12th 2010, 01:16 PM
  4. Triangle Inequality
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Feb 9th 2010, 03:16 PM
  5. Triangle Inequality
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: Sep 25th 2009, 03:06 AM

/mathhelpforum @mathhelpforum