1. ## 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.

2. ## Re: Triangle Inequality

Originally Posted by kastamonu
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.

3. ## 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.

4. ## Re: Triangle Inequality

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

so the problem has only two solutions $\displaystyle a=6$ and $\displaystyle a=7$

5. ## 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 $a=7$.

7. ## Re: Triangle Inequality

Originally Posted by kastamonu
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."

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.

8. ## 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.

9. ## Re: Triangle Inequality

Originally Posted by kastamonu
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.

10. ## Re: Triangle Inequality

2 solutions ONLY:
a-b-c
6-3.5-8.5
7-2.5-7.5