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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]



Answer :

GinnyAnswer
To determine the range of possible values for the third side of a triangle given two sides measuring 10 cm and 16 cm, we need to apply 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]

In our case, we have two known sides:
- [tex]\(a = 10 \text{ cm}\)[/tex]
- [tex]\(b = 16 \text{ cm}\)[/tex]

Let's denote the unknown third side by [tex]\(x\)[/tex]. We need to apply the triangle inequality theorem to find the constraints for [tex]\(x\)[/tex].

1. From [tex]\(a + b > x\)[/tex]:
[tex]\[10 + 16 > x \implies 26 > x \implies x < 26\][/tex]

2. From [tex]\(a + x > b\)[/tex]:
[tex]\[10 + x > 16 \implies x > 16 - 10 \implies x > 6\][/tex]

3. From [tex]\(x + b > a\)[/tex]:
[tex]\[x + 16 > 10\][/tex]
This inequality will always be true for any positive value of [tex]\(x\)[/tex], so it does not provide any additional constraints.

Combining the inequalities we obtained:
[tex]\[6 < x < 26\][/tex]

Hence, the range of possible values for the third side [tex]\(x\)[/tex] must be greater than 6 and less than 26. Therefore, the most accurate description of the range of possible values for the third side of this triangle is:

[tex]\[6 < x < 26\][/tex]

So, the correct answer is:
[tex]\[6

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