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 < 13.5, x > 18.9 )
• ( 13.5 < x < 18.9 )
• ( x < 6, x > 26 )
• ( 6 < x < 26 )

Explanation:

Step1: Apply 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. Let the sides be \(a = 10\), \(b = 16\), and \(c=x\).

First, \(a + b>c\): \(10 + 16>x\) \(\Rightarrow\) \(x < 26\).

Second, \(a + c>b\): \(10 + x>16\) \(\Rightarrow\) \(x>16 - 10\) \(\Rightarrow\) \(x > 6\).

Third, \(b + c>a\): \(16 + x>10\) (this is always true for positive \(x\) since \(x>0\) and \(16 + x\) will always be greater than \(10\)).

Combining the valid inequalities from the triangle inequality theorem, we get \(6

Answer:

\(6 < x < 26\) (assuming the last option in the image is \(6 < x < 26\), as per the triangle inequality and the steps above)