How many different isosceles triangles are possible if the sides must have whole number lengths and the perimeter must be ninety-three inches?
In any triangle whose sides have measures a,b, and c,
the triangular inequality tells us:
a + b > c
a + c > b
b + c > a
Since the triangles here have two sides with the same
measure, we will suppose that a = b and the above becomes
a + a > c
a + c > a
a + c > a
or
2a > c
c > 0
c > 0
So we have
1) 2a > c > 0
Since the perimeter is 93,
we have
a + a + c = 93
2) 2a + c = 93
Solve 2) for c,
c = 93 - 2a
Substitute in 1)
1) 2a > 93 - 2a
4a > 93
a > 23.25
and since a is an integer
a > 23
Solve 2) for 2a
2a = 93 - c
Substitute 93 - c for 2a in 1)
93 - c > c
93 > 2c
46.5 > c
and since c is an integer,
47 > c
or c < 47
So we have
a > 23
c < 47
Let p be the positive integer amount that a is greater than 23
Let q be the positive integer amount that a is less than 47
So
a = 23 + p
and
c = 47 - q
Substituting those into 2)
2a + c = 93
2(23 + p) + (47 - q) = 93
46 + 2p + 47 - q = 93
93 + 2p - q = 93
2p - q = 0
2p = q
So now we have
a = 23+p
and
c = 47-2p
Since c > 0
47 - 2p > 0
-2p > -47
p < 23.5
and since p is an integer
p < 24
And p > 0 so
0 < p < 24
So there are 23 values p can take on, any of the integers
which are greater than 0 and less than 24. That's 1 through
23, so there are 23 possible isoceles triangles with integer
sides that have perimeter 93.
Here they all are:
p a = 23+p b = a c = 47-2p
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1. 24 24 45
2. 25 25 43
3. 26 26 41
4. 27 27 39
5. 28 28 37
6. 29 29 35
7. 30 30 33
8. 31 31 31
9. 32 32 29
10. 33 33 27
11. 34 34 25
12. 35 35 23
13. 36 36 21
14. 37 37 19
15. 38 38 17
16. 39 39 15
17. 40 40 13
18. 41 41 11
19. 42 42 9
20. 43 43 7
21. 44 44 5
22. 45 45 3
23. 46 46 1
Edwin
