Question

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? x < 12.5, x > 18.9 12.5 < x < 18.9 x < 6, x > 26 6 < x < 26

Explanation:

Step1: Triangle - side relationship

For a triangle with sides \(a = 10\), \(b = 16\), and \(c=x\), the general triangle - side relationship is \(|a - b|\lt c\lt a + b\). So \(|10 - 16|\lt x\lt10 + 16\), which simplifies to \(6\lt x\lt26\). But since it's an acute triangle, we use the Pythagorean - related inequalities.
If \(x\) is the longest side (\(x\geq16\)), then \(a^{2}+b^{2}>c^{2}\). Substitute \(a = 10\), \(b = 16\), we get \(10^{2}+16^{2}>x^{2}\), \(100 + 256>x^{2}\), \(x^{2}<356\), \(x<\sqrt{356}\approx18.9\).
If \(16\) is the longest side (\(x<16\)), then \(a^{2}+x^{2}>b^{2}\). Substitute \(a = 10\), \(b = 16\), we get \(10^{2}+x^{2}>16^{2}\), \(x^{2}>16^{2}-10^{2}\), \(x^{2}>256 - 100\), \(x^{2}>156\), \(x>\sqrt{156}\approx12.5\).

Answer:

\(12.5\lt x\lt18.9\)