My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
March 22nd, 2014, 07:26 PM   #1
Senior Member
 
Joined: Apr 2012

Posts: 799
Thanks: 1

which one is the lengthiest


with side length :

we have :


now determine among :

which one is the lengthiest
Albert.Teng is offline  
 
March 22nd, 2014, 08:13 PM   #2
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,655
Thanks: 2633

Math Focus: Mainly analysis and algebra
Re: which one is the lengthiest

You want to express b in terms of a only and c in terms of a only. You can do this by eliminating b and c in turn from the equations in the same manner that you would to solve a system of simultaneous equations. You should then be able to order them appropriately.

I made for all values of and when .
v8archie is offline  
March 22nd, 2014, 08:23 PM   #3
Math Team
 
Joined: Oct 2011
From: Ottawa Ontario, Canada

Posts: 14,412
Thanks: 1024

Re: which one is the lengthiest

c.

Also, if a,b,c are integers, then b>a except for one case: (a,b,c) = (5,3,7)
Denis is offline  
March 22nd, 2014, 09:26 PM   #4
Senior Member
 
Joined: Mar 2014

Posts: 112
Thanks: 8

0 = aČ - 4c + 3 i.e. 4c = aČ + 3. Hence c = (aČ + 3)/4.
0 = aČ - 2a - 4b - 3 i.e. 4b = aČ - 2a - 3 = (a - 3)(a + 1). Hence b = (a - 3)(a + 1)/4.
Since b > 0 is correct, a > 3 must also be correct. If we use the solution Denis gave we are done.
the john is offline  
March 24th, 2014, 09:14 PM   #5
Math Team
 
Joined: Oct 2011
From: Ottawa Ontario, Canada

Posts: 14,412
Thanks: 1024

Re: which one is the lengthiest

Quote:
Originally Posted by Albert.Teng
Was diddling with this "Albert Special" during the hockey game...

Above equation can be rewritten as:
2c - 2b = a + 3 [1]
So evident that c > b (since a,b,c are all > 0, being triangle sides)

Can't find "as simple" a way to show c > a...
Denis is offline  
March 25th, 2014, 11:24 AM   #6
Math Team
 
Joined: Oct 2011
From: Ottawa Ontario, Canada

Posts: 14,412
Thanks: 1024

Re: which one is the lengthiest

Albert, something for your "useless information(!)" file;
here's the first 8 integer solutions:
Code:
n    a    b    c
1    5    3    7
2    7    8   13
3    9   15   21
4   11   24   31
5   13   35   43
6   15   48   57
7   17   63   73
8   19   80   91
a = 2(n - 1) + 5
b = n(n + 2)
c = n^2 + 3n + 3

You're welcome; no charge :P
Denis is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
lengthiest



Thread Tools
Display Modes






Copyright © 2019 My Math Forum. All rights reserved.