+1940 All the sides of a triangle have integer length. The perimeter of the triangle is 10, and the triangle is scalene. How many such non-congruent triangles are there?
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All the sides of a triangle have integer length. The perimeter of the triangle is 10, and the triangle is scalene. How many such non-congruent triangles are there?
Consider one side of the triangle, call it a.
If a = 1, then b+c must equal 9. 2+7, 3+6, 4+5 all violate the Triangle Inequality Theorem.
a = 2, then b+c must equal 8. 1+7, 3+5 violate Triangle Inequality.
a = 3, then b+c must equal 7. 1+6, 2+5 violate the theorem.
Try it for all lengths of a. There are zero scalene triangles that satisfy the conditions.
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