An acute triangle has two sides measuring 8 cm and 10 cm. What is the best representation of the possible range of values for the third side, $s$?

A. $2
B. $6
C. $s<2$ or $s>18$
D. $s<6$ or $s>12.8$

Apply the triangle inequality theorem to establish a preliminary range for the third side: 2 < s < 18 .
Use the acute triangle condition, considering both cases where 10 is the longest side ( 6"> s > 6 ) and where s is the longest side ( s < 164 ​ ≈ 12.8 ).
Combine the triangle inequality and acute triangle conditions to refine the range.
The final range for the third side is 6 < s < 12.8 .

Explanation

Problem Analysis We are given an acute triangle with two sides measuring 8 cm and 10 cm. We need to find the possible range of values for the third side, s .

Triangle Inequality First, we apply the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side. This gives us two inequalities:


s"> 8 + 10 > s and 10"> 8 + s > 10 and 8"> 10 + s > 8
From s"> 8 + 10 > s , we get s < 18 .
From 10"> 8 + s > 10 , we get 2"> s > 2 .
From 8"> 10 + s > 8 , we get -2"> s > − 2 (which is trivial since s must be positive).
Combining these, we have 2 < s < 18 .

Acute Triangle Condition Since the triangle is acute, the square of the longest side must be less than the sum of the squares of the other two sides. We consider two cases:

Case 1: 10 is the longest side. Then, 1 0 2 < 8 2 + s 2 , which simplifies to 100 < 64 + s 2 . This gives 36"> s 2 > 36 , so 6"> s > 6 .
Case 2: s is the longest side. Then, s 2 < 8 2 + 1 0 2 , which simplifies to s 2 < 64 + 100 , so s 2 < 164 . This means s < 164 ​ ≈ 12.806 .

Combining Conditions Combining the results from the triangle inequality ( 2 < s < 18 ) and the acute triangle condition ( 6"> s > 6 and s < 164 ​ ≈ 12.806 ), we find the range of possible values for s to be 6 < s < 12.806 . Since we are looking for the best representation, we can approximate 12.806 as 12.8 .

Final Answer Therefore, the best representation of the possible range of values for the third side s is 6 < s < 12.8 .


Examples
Imagine you are building a triangular garden bed and two sides must be 8 meters and 10 meters long. Knowing the possible range of the third side ensures that your garden bed is not only buildable (triangle inequality) but also has angles less than 90 degrees (acute triangle). This knowledge helps you design a garden that is both structurally sound and aesthetically pleasing.

Answered by GinnyAnswer | 2025-07-06