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Given 10 and 28 as the lengths of two sides of a triangle, find the range of values for the third side.

18 < x < 38

Enter the number that belongs in the box.

Sagot :

GinnyAnswer
To solve this problem, we need to determine the possible range for the length of the third side of a triangle when the other two sides measure 10 and 28 units. We'll use the triangle inequality theorem, which states the following:

1. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
2. The difference between the lengths of any two sides of a triangle must be less than the length of the third side.

Let's denote the three sides of the triangle as [tex]\( a = 10 \)[/tex], [tex]\( b = 28 \)[/tex], and [tex]\( c \)[/tex] (where [tex]\( c \)[/tex] is the unknown third side).

According to the triangle inequality theorem, we need to satisfy these inequalities:

1. [tex]\( a + b > c \)[/tex] \rightarrow [tex]\( 10 + 28 > c \)[/tex] \rightarrow [tex]\( 38 > c \)[/tex]
2. [tex]\( a + c > b \)[/tex] \rightarrow [tex]\( 10 + c > 28 \)[/tex] \rightarrow [tex]\( c > 18 \)[/tex]
3. [tex]\( b + c > a \)[/tex] \rightarrow [tex]\( 28 + c > 10 \)[/tex] \rightarrow [tex]\( c > -18 \)[/tex] (this condition is always true and does not impose any additional limitations).

So, taking into account the inequalities from steps 1 and 2:

- [tex]\( c > 18 \)[/tex]
- [tex]\( c < 38 \)[/tex]

Therefore, the range of values for [tex]\( c \)[/tex] is [tex]\( 18 < c < 38 \)[/tex].

This means that the range for the third side [tex]\( c \)[/tex] is between 19 and 37 inclusively.

The value that belongs in the green box is

[tex]\[ 37 \][/tex]
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