Part A
Let,
m = missing side of the triangle
It turns out by the triangle inequality theorem that if we know two sides of a triangle, then the third side m is constrained by the compound inequality:
b-a < m < b+a
The value of b is larger than 'a'; unless a = b.
In this case, a = 13 and b = 19 are the two known sides, so...
b-a < m < b+a
19-13 < m < 19+13
6 < m < 32
The missing side length is between 6 and 32 units.
The missing side length cannot be 6 units long, or else a triangle won't form (we'll get a straight line instead)
The missing side length cannot be 32 units long, or else a triangle won't form (we'll get a straight line instead)
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Part B
Same thing as before but now a = 14 and b = 15
m = missing side
b-a < m < b+a
15-14 < m < 15+14
1 < m < 29
The missing side length is between 1 and 29 units.
The missing side length cannot be 1 unit long, or else a triangle won't form (we'll get a straight line instead)
The missing side length cannot be 29 units long, or else a triangle won't form (we'll get a straight line instead)
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Part C
a & b aren't known numerically so we just leave it as
b-a < m < b+a
The missing side m is between b-a units and b+a units
m cannot equal b-a or else we'll form a straight line (and not a triangle)
m cannot equal b+a or else we'll form a straight line (and not a triangle)
