In the triangle shown, for \(\angle A\) to be the largest angle of the triangle, it must be that \(m < x < n\). What is the least possible value of \(n - m\), expressed as a common fraction?

Guest Oct 12, 2018

#1**0 **

Did you say thank you to anyone, or appologize for the incorrect posting on the original.

I do not think I ever witness either of those events.

What have you changed anyway?

Perhaps you could give us a link to the original post so that we can see what you have changed AND so that we can see your good manners displayed too.

Melody
Oct 12, 2018

#15**+1 **

I'm very sorry Melody! Even though you took time out of your day to help me with the problem multiple times, I completely failed to thank you. I also never apologized to you for reposting the question without properly apologizing that I wasted everyone's time on a problem flawed by my own carelessness. Once again, I am so sorry and thank you for being patient with me, despite all of my wrongdoings!

Guest Oct 13, 2018

#3**0 **

Thanks very much guest#2 for digging that post out.

Well i can see no thank you or interaction from out poster at all.

I also have not noticed and difference in the wording.

So unless demonstrated otherwise, ... well you can all fill in this blank .......

Perhaps you, guest#1 would like to point out this 'crucial typo'

Melody
Oct 12, 2018

#4**+1 **

I think he changed the wording of the question, here's how i interpreted it:

Let C be the set of all real numbers x such that the largest angle of the triangle in the picture is A. We need to find sup(C)-inf(C).

Also I think he responded in his previous post too.

Guest Oct 12, 2018

edited by
Guest
Oct 12, 2018

#5**+1 **

Here are all the restrictions we can place on x (that I can think of!):

∠A is the largest angle, so the side opposite ∠A is the largest side, so...

x + 9 > 3x

9 > 2x

x < \(\frac92\)

x + 9 > x + 4

9 > 4 (Any value of x makes this inequality true.)

The sum of any two sides of a triangle must be greater than the third side, so...

(3x) + (x + 4) > x + 9

4x + 4 > x + 9

3x > 5

x > \(\frac53\)

(3x) + (x + 9) > x + 4

4x + 9 > x + 4

3x > -5

x > -\(\frac53\)

(x + 9) + (x + 4) > 3x

2x + 13 > 3x

x < 13

For A to be the largest angle, it must be that: x < \(\frac92\) and 9 > 4 and x > \(\frac53\) and x > -\(\frac53\) and x < 13

We can simplify that to say...

For A to be the largest angle, it must be that: x < \(\frac92\) and x > \(\frac53\)

In other words...

For A to be the largest angle, it must be that: \(\frac53\) < x < \(\frac92\)

And n - m = \(\frac92\) - \(\frac53\) = \(\frac{27}{6}\) - \(\frac{10}{6}\) = \(\frac{17}{6}\)

But it is confusing that it says, "What is the *least* possible value of n - m," when it seems like there is only one possible value of n - m .

hectictar
Oct 12, 2018

#6**0 **

"In the triangle shown, for A to be the largest angle it must be that m

You found the "optimal" values for m and n- the largest value for m that satisfies the statement and the smallest value for n that satisfies the statement. It is also correct to say that x must be between 0 and 100 (0

Guest Oct 12, 2018

edited by
Guest
Oct 12, 2018

edited by Guest Oct 12, 2018

edited by Guest Oct 12, 2018

#7**0 **

"In the triangle shown, for A to be the largest angle it must be that m

You found the "optimal" values for m and n- the largest value for m that satisfies the statement and the smallest value for n that satisfies the statement. It is also correct to say that x must be between 0 and 100 (0

Guest Oct 12, 2018

#8**+1 **

Ok for some reason I can't edit my answer

EDIT: testing

"In the triangle shown, for A to be the largest angle it must be that m

You found the "optimal" values for m and n- the largest value for m that satisfies the statement and the smallest value for n that satisfies the statement. It is also correct to say that x must be between 0 and 100, which is why the guest asked for the "least possible value of n-m". In other words, we are asked to find the difference between the least upper bound and the largest lower bound of the values of x that satisfy the statement. You found the least upper bound (9/2) and the largest lower bound (5/3) so by finding the difference between them you found the answer.

Guest Oct 12, 2018

edited by
Guest
Oct 12, 2018

edited by Guest Oct 12, 2018

edited by Guest Oct 12, 2018

edited by Guest Oct 12, 2018

edited by Guest Oct 12, 2018

#9**+1 **

Hmmmm....ohh...I think you are right

This is a true statement:

For ∠A to be the largest angle, it must be that 0 < x < 100

So there is more than one possible value of n - m .

( And it is also true that if 2.9999 < x < 3 , then ∠A is the largest angle,

But it is not true that it *must *be that 2.9999 < x < 3 for ∠A to be the largest angle. )

hectictar
Oct 12, 2018

#11**+1 **

Dear Melody, Hectitar, and all of the Guests who helped me on my problems,

Hello, I am Guest #1. I am extremely sorry for giving everyone a flawed question, and even more so, for not thanking anyone for their efforts. Even though you guys used so much of your time to try and help me solve the problem, I have completely failed in saying thank you to everyone in the original post. I have also failed to properly apologize in this reposting of the question for my error. I am truly sorry and I will make sure to never be this ungrateful and rude again. Thank you and sorry to all of you, who have been so patient even though I have demonstrated such a poor attitude.

Extremely sorry,

Guest #1

Guest Oct 13, 2018