To determine the range of possible lengths for the third side of a triangle given that the other two sides are 200 units and 300 units, we will use the triangle inequality theorem. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
Let's denote the sides of the triangle as [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]. For our specific problem, we have:
- [tex]\( a = 200 \)[/tex]
- [tex]\( b = 300 \)[/tex]
- Let [tex]\( c = x \)[/tex], where [tex]\( x \)[/tex] is the length of the third side that we need to determine.
We need to satisfy the following three inequalities to ensure a valid triangle:
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
1. [tex]\( 200 + 300 > x \)[/tex]
2. [tex]\( 200 + x > 300 \)[/tex]
3. [tex]\( 300 + x > 200 \)[/tex]
Simplifying these inequalities, we get:
1. [tex]\( 500 > x \)[/tex]
2. [tex]\( x > 100 \)[/tex]
3. [tex]\( x > -100 \)[/tex] (This last inequality is redundant since [tex]\( x \)[/tex] must be positive in the context of triangle side lengths.)
Therefore, combining the valid inequalities, we can write the compound inequality for [tex]\( x \)[/tex] as:
[tex]\[ 100 < x < 500 \][/tex]
This is the range of possible lengths for the third side of the triangle when the other two sides are 200 units and 300 units.
