1) In acute triangle ABC, we know AB = 7, BC = 8, and that CA is the shortest side. What is the smallest possible integer value of CA?

2) Two diagonals of a parallelogram have lengths 6 and 8. What is the largest possible length of the shortest side of the parallelogram?

3) Two sides of an acute triangle are 8 and 15. How many possible lengths are there for the third side if it is an integer?

4) We can find an acute triangle with the three altitude lengths 1, 2, and h, if and only if (h^2 )belongs to interval (p, q) Find (p, q).

I had some trouble with these problems. I got (13,14,15,16) for #3 but it was incorrect.

Thanks!

Rudram592 Jan 26, 2020

#1**+1 **

1) For this problem, you would use a formula similar to the Pythag theorem. The pythag theorem says a^2 + b^2 = c^2 for right triangles, so if a triangle is acute, a^2 + b^2 > c^2. Likewise for obtuse triangles, a^2 + b^2 < c^2.

Use this and brute force to find the answer.

Hope this helps,

- PM

PartialMathematician Jan 26, 2020

#2**+2 **

2) Think about it. The question says to find the "largest possible length of the shortest side". Well if you go too big, the "shortest side" becomes the "longest side". What if there was no "shortest side" and "longest side"?

- PM

PartialMathematician Jan 26, 2020

#4**0 **

OK I thought about it. You obviously hope this answer doesn’t help, because your answer is pure B U L L S H I T!

Guest Jan 26, 2020

#6**+2 **

Lol I thought you were smart enough to understand... The parallelogram is a rhombus.

rhom - bus

/ˈrämbəs/

noun

a parallelogram with opposite equal acute angles, opposite equal obtuse angles, and four equal sides.

any parallelogram with equal sides, including a square.

The largest possible length of the shortest side is 5.

PartialMathematician
Jan 26, 2020

#7**+2 **

And good catch, guest!

I hope the above solution helps.

Hope this helps,

- PM

PartialMathematician
Jan 26, 2020

#8**0 **

**A parallelogram is NOT a rhombus, you DUMB F U C K F A C E!**

The largest possible value for the shortest side is < 3

Guest Jan 26, 2020

#10**+3 **

I am very sorry guest (for your mother) but you must be describing yourself.

A Rhombus IS a special type of PARALLELOGRAM

**PartialMathematician is correct.**

Here is a pic that I could not resist drawing.

Melody
Jan 26, 2020

#11**0 **

*Melody Wrote: A Rhombus IS a special type of PARALLELOGRAM. PartialMathematician is correct.*

OK that’s true, **but that is NOT what PartialMathematician wrote**.

He wrote, “**The parallelogram is a rhombus**.” That is not the case here nor anywhere else.

You can know the parallelogram is not a rhombus because the question states** the two diagonals of a parallelogram have lengths 6 and 8**. If this was a rhombus, they would be the same length.

The diagonals bisect each other forming 4 triangles that have sides of 3 and 4. Then applying the triangle inequality z ≤ x + y that includes the *degenerate triangle* where z = x + y and appears as a line with zero area, then the **unknown side (z) **≤** side (x) + side (y) therefore z is longer than either x or y. Then x = 3 is the longest the shortest side can be, or x < 3 is the longest the shortest side can be excluding the degenerate triangle with zero area. **

This proves PartialMathematician is **wrong and he is still a dumb F u c k F a c e and he would be one even if he were right. **

I like your moving parallelogram picture. It’s a rhombus (special case) when the points are on the y axis.

Melody Also Wrote:* I am very sorry guest (for your mother) but you must be describing yourself.*

I might be a F u c k F a c e, but my mother never noticed or at least never said so. There is probably some kind of genetic code that prevents mothers from seeing these kinds of defects in their offspring. My father didn’t have this genetic code because he sometimes referred to my brother as F u c k F a c e. I was never sure if it was a term of endearment, or if my father thought my brother wasn’t his child. We did look different, so I don’t know. I looked different from my sister too. One main difference was I had much bigger b o o b s, and this p i s s e d her off to no end. She would call me F u c k F a c e when I pointed this out.

Guest Jan 26, 2020

edited by
Guest
Jan 26, 2020

#12**+2 **

**You are making a total fool of yourself guest.**

**I can see why you want to be anonymous.**

As you have stated, PartialMathematician wrote

“The parallelogram is a rhombus.”

And you responded with

"That is not the case here nor anywhere else."

**He clearly meant that this Parallelogram is a rhombus and he is absolutely correct.**

You seem to think that a rhombus and a square are the same thing!

A square is a special case of a rhombus, that is true enough, but a rhombus does not usually have equal diagonals.

This is the exact rhombus that meets the necessary description.

Melody
Jan 26, 2020

#13**0 **

*You are making a total fool of yourself guest.*

*I can see why you want to be anonymous.*

Are you sure that I’m making a **total fool** of myself? I think I’m making only a **partial**-fool of myself.

If you can see why I want to be **anonymous, **would you please tell me? It’s not because I’m worried about making a **partial **fool of myself. That should be obvious, because I had no idea I was doing that until you posted the graph that doesn’t cause **partial** motion-sickness.

I may make an account. I have a **partial** shortlist of user names.

**PartialFool**

**PartialIdot **

**PartialDumbFuck**

**PartialFuckFace. **

Are you **partial** to one of them?

Guest Jan 26, 2020

#16**0 **

How about smartmouthfool?

But I will think on it if you want.

You have definitely made a total fool of yourself. Not just a partial one.

ok I get it. PartialMathematician, PartialFool....

But it still does not fit for you.

Maybe

PartialMathematiciansmoron but PartialMathematician may not want to be linked to you for ever.

PartialMathematiciansidiot.

PartialMathematiciansfool. That one is not too bad.

PartialMsFool Maybe that one. Yea that seems ok.

PartialMsTotalFool

Actually the fool in a yesteryear King's court was the only person who could get away with criticising the King, whether he was a idiot or not. He usually kept his head, in a literal sense, when others did not. So maybe being PM's fool might appeal to you. Everyone else on the forum will either know or not care where the name comes from but you can live in a make believe world thinking you are a witty intelligent fool.

Melody
Jan 26, 2020

#9**+1 **

1) In acute triangle ABC, we know AB = 7, BC = 8, and that CA is the shortest side. What is the smallest possible integer value of CA?

The smallest possible integer value of CA is 4

3) Two sides of an acute triangle are 8 and 15. How many possible lengths are there for the third side if it is an integer?

**There are only 4 possible lengths and they are: 8, 9, 10 and 11 **

Dragan Jan 26, 2020

#14**+1 **

Guest, do you seriously know how you are talking to. If you really had some appreciation for learning, maybe you'd see our answerers in a better light. It's not the answers that matter, it's the process.

DragonLord Jan 26, 2020