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An acute triangle has sides measuring 10 cm and 16 cm. The length of the third side is unknown.

Which best describes the range of possible values for the third side of the triangle?

A. [tex]\( x \ \textless \ 12.5 \)[/tex] or [tex]\( x \ \textgreater \ 18.9 \)[/tex]
B. [tex]\( 12.5 \ \textless \ x \ \textless \ 18.9 \)[/tex]
C. [tex]\( x \ \textless \ 6 \)[/tex] or [tex]\( x \ \textgreater \ 26 \)[/tex]
D. [tex]\( 6 \ \textless \ x \ \textless \ 26 \)[/tex]

Sagot :

GinnyAnsweridn
To determine the range of possible values for the third side of a triangle where the other two sides are 10 cm and 16 cm, you need to use the triangle inequality theorem. This theorem states that for any triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:

1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]

For this specific triangle:

1. Let [tex]\( a = 10 \)[/tex], [tex]\( b = 16 \)[/tex], and [tex]\( c \)[/tex] be the unknown side.
2. We apply the inequalities as follows:

- [tex]\( 10 + 16 > c \implies 26 > c \)[/tex]
- [tex]\( 10 + c > 16 \implies c > 6 \)[/tex]
- [tex]\( 16 + c > 10 \implies this is always true since \( c \)[/tex] is positive.

Combining the first two conditions, we get:
[tex]\[ 6 < c < 26 \][/tex]

Therefore, the length of the third side lies between 6 cm and 26 cm.

The correct answer is:
[tex]\[ 6 < x < 26 \][/tex]
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