Question

the longest side of an isosceles obtuse triangle measures 20 centimeters. the other two side lengths are congruent but unknown. what is the greatest possible whole - number value of the congruent side lengths? 9 cm 10 cm 14 cm 15 cm

Explanation:

Step1: Apply triangle inequality

For a triangle with sides \(a\), \(a\), and \(c = 20\) (\(a\) is the length of the congruent sides), by triangle inequality \(a+a>20\), i.e., \(2a > 20\), \(a>10\).

Step2: Use the condition for obtuse - angled triangle

If the triangle is obtuse - angled with the longest side \(c = 20\), then by the law of cosines \(c^{2}>a^{2}+a^{2}\) (since the angle opposite the longest side is obtuse). Substitute \(c = 20\) into \(c^{2}>2a^{2}\), we get \(20^{2}>2a^{2}\), \(400>2a^{2}\), \(a^{2}<200\), \(a < \sqrt{200}\approx14.14\).

Answer:

\(14\space cm\)