How many different isosceles triangles are possible if the sides must have whole number lengths and the perimeter must be 93 inches?

Answer 1

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 ---------------------------------- 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