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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 an acute triangle with given sides of 10 cm and 16 cm, we can use the triangle inequality theorem.

The triangle inequality theorem states:
1. The sum of any two sides of a triangle must be greater than the third side.
2. The difference of any two sides of a triangle must be less than the third side.

Let's break this down step by step for the given sides of 10 cm and 16 cm:

1. The third side must be greater than the difference of the given sides:
[tex]\[ \text{Difference} = |16 - 10| = 6 \][/tex]
Therefore, the third side must be greater than 6 cm.

2. The third side must be less than the sum of the given sides:
[tex]\[ \text{Sum} = 10 + 16 = 26 \][/tex]
Therefore, the third side must be less than 26 cm.

Combining these two results, the range of possible values for the third side, [tex]\( x \)[/tex], is:
[tex]\[ 6 < x < 26 \][/tex]

So, the correct answer is:
[tex]\[ 6 < x < 26 \][/tex]
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