Two sides of an integer sided triangle have lengths 18 and
m. If n represents 35 values of an integer such that each of the triangles having lengths 18,
m, and n is non-degenerate.
Determine the total number of values that m can assume.
In general, if two sides of a triangle are a and b, a <= b, you can picture a line segment of length b, and a circle of radius a centered at one of the endpoints of b. The 3rd vertex of a triangle can be anywhere on that circle, except if the three vertices are all colinear.
The length of side c can be anywhere from (b-a) < c < (b+a).
In the case of integer lengths, the third side can be any of 2a-1 integer lengths. (a <= b).
If the longer side is 18 and the shorter side is m, there can be 2m-1 triangles; none of these can be 35.
If the shorter side is 18 and the longer side is m, there can be 2*18-1 = 35 triangles. So for any value of m > 18, there can be 35 triangles.
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Posted by Larry
on 2024-08-14 11:58:11 |
Solution