# Show all the triangles that can be made each using 24 matches Also show all isoscles triangles that can be made with 24 matches?

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If you have 24 matches and you want to see how many different triangles you can make with them, you can think of each match as one unit and determine how many triangles can possibly have a perimeter of 24 units. The sides of each triangle are made by lining up a certain number of matches, so we know that we can only have whole number lengths for the triangles' sides. Recall that for any triangle, the sum of any two sides is greater than the length of the third side. This means that if the 3 sides of a triangle are called a, b, & c:

a + b > c

b + c > a

a + c > b

We also know that a + b + c = 24. Considering all of these equations together, we can determine that:

a < 12

b < 12

c < 12

because if any one side were 12 or more matches long, the other 2 sides could not add together to be greater than 12. From here you can just go through possible combinations to determine lengths for a, b, & c which add up to 24, and in which no one side is greater than or equal to 12. The possibilities are:

(8,8,8)
(7,6,11)
(7,8,9)
(6,9,9)
(6,10,8)
(9,5,10)
(5,11,8)
(9,4,11)
(10,7,7)
(4,10,10)
(3,10,11)
(2,11,11)

So there are 12 possibile triangles with a perimeter of 24, meaning you can use 24 matches to make 12 different triangles. Isosceles triangles are ones which have exactly 2 sides of equal length, so we can count how many triangles have this property:

(6,9,9)
(10,7,7)
(4,10,10)
(2,11,11)

So there are 4 ways to make an isosceles triangle.

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There is one other isosceles triangle, (8,8,8). Even though it is an equilateral triangle, it is also isosceles - which has at least 2 sides the same, therefore (8,8,8) will work. :P