SOLUTION: Sam is studying isosceles triangles. He wants to know how many isosceles triangles exist with a perimeter of 99 inches and side lengths are positive whole numbers. What are the len

Algebra ->  Customizable Word Problem Solvers  -> Geometry -> SOLUTION: Sam is studying isosceles triangles. He wants to know how many isosceles triangles exist with a perimeter of 99 inches and side lengths are positive whole numbers. What are the len      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 535964: Sam is studying isosceles triangles. He wants to know how many isosceles triangles exist with a perimeter of 99 inches and side lengths are positive whole numbers. What are the lengths of the sides of these triangles? How do you know when you've found them all?
Found 3 solutions by edjones, richwmiller, Edwin McCravy:
Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
There are 49 possible triangles. Each of the equal sides can be from 1 to 49 inclusive.
.
Ed

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
let the equal sides be x and the third side be y
2x+y=99
let y be 1 through 97
where 2x>y
since the sum of any two sides must be greater than the third side.


Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
Let the two equal sides have length a=b and the base have length c

The perimeter = a+b+c = a+a+c = 2a+c = 99

By the triangle inequality we always have a + c > a true so the only
triangular inequality we have to be concerned about is

a + a > c 

or

2a > c

and

c > 0

So putting everything we have together:

(1)     2a+c = 99
(2)       2a > c
(3)        c > 0

Solve (1) for c

(1)     2a+c = 99
(4)        c = 99 - 2a  

Substitute (4) into (2)

(2)       2a > c
          2a > 99 - 2a
          4a > 99
(5)        a > 24.75

Substitute (4) into (3)

(3)        c > 0
     99 - 2a > 0
         -2a > -99
(6)        a < 49.5

Putting (5) and (6) together

        24.75 < a < 49.75

So a can be any integer from 25 to 49, inclusive.

There are 49 integers from 1 thru 49.
We subtract the number of integers 34 or less, and there are 24
of them, so the number of triangles satisfying the given
conditions is 49 - 24 or 25.

That's the answer, 25.

Here they all are, where "a" can be any integer from 25 
through 49, inclusive.

     a  b  c
 1.  25 25 49
 2.  26 26 47
 3.  27 27 45
 4.  28 28 43
 5.  29 29 41
 6.  30 30 39
 7.  31 31 37
 8.  32 32 35
 9.  33 33 33
10.  34 34 31
11.  35 35 29
12.  36 36 27
13.  37 37 25
14.  38 38 23
15.  39 39 21
16.  40 40 19
17.  41 41 17
18.  42 42 15
19.  43 43 13
20.  44 44 11
21.  45 45 9
22.  46 46 7
23.  47 47 5
24.  48 48 3
25.  49 49 1

Edwin