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

Explanation:

Step1: Apply triangle - inequality theorem

For a triangle with side lengths \(a\), \(b\), and \(c\), the following inequalities must hold: \(|a - b|\lt c\lt a + b\). Here \(a = 10\) and \(b = 16\). First, calculate \(|10 - 16|=6\) and \(10 + 16 = 26\). So, \(6\lt x\lt26\).

Step2: Consider the condition for an acute - angled triangle

For an acute - angled triangle with side lengths \(a\), \(b\), and \(c\) (where \(c\) is the longest side), \(a^{2}+b^{2}>c^{2}\).
If \(x\) is the longest side (\(x\geq16\)), then \(10^{2}+16^{2}>x^{2}\), i.e., \(100 + 256>x^{2}\), \(x^{2}<356\), \(x < \sqrt{356}\approx18.9\).
If \(16\) is the longest side (\(x < 16\)), then \(10^{2}+x^{2}>16^{2}\), \(100+x^{2}>256\), \(x^{2}>156\), \(x>\sqrt{156}\approx12.5\). So, \(12.5\lt x\lt18.9\).

Answer:

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