Question

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?
○ $2 < s < 18$
○ $6 < s < 12.8$
○ $s < 2$ or $s > 18$
○ $s < 6$ or $s > 12.8$

Explanation:

Step1: Recall Triangle Inequality Theorem

For any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. So, for sides \(a = 8\), \(b = 10\), and \(c = s\), we have:
\( |a - b| < s < a + b \)
Substituting \(a = 8\) and \(b = 10\), we get \( |8 - 10| < s < 8 + 10 \), which simplifies to \( 2 < s < 18 \). But since it's an acute triangle, we need to use the Pythagorean theorem for acute triangles.

Step2: Case 1 - s is the longest side

If \(s\) is the longest side, then for the triangle to be acute, \(a^2 + b^2 > s^2\). So \(8^2 + 10^2 > s^2\), \(64 + 100 > s^2\), \(164 > s^2\), so \(s < \sqrt{164}\approx12.8\).

Step3: Case 2 - 10 is the longest side

If 10 is the longest side, then \(a^2 + s^2 > b^2\). So \(8^2 + s^2 > 10^2\), \(64 + s^2 > 100\), \(s^2 > 36\), so \(s > 6\) (since side length is positive).

Combining the two cases and the triangle inequality, we get \(6 < s < 12.8\).

Answer:

B. \(6 < s < 12.8\)