Question

list the sides of the triangle in order from shortest to longest. choose the correct order of the sides from shortest to longest. a. \\( \overline { o m } , \overline { n o } , \overline { n m } \\) b. \\( \overline { n m } , \overline { n o } , \overline { o m } \\) c. \\( \overline { n m } , \overline { o m } , \overline { n o } \\) d. \\( \overline { o m } , \overline { n m } , \overline { n o } \\) e. \\( \overline { n o } , \overline { n m } , \overline { o m } \\) f. \\( \overline { n o } , \overline { o m } , \overline { n m } \\)

Explanation:

Step1: Find the measure of angle \(O\)

The sum of angles in a triangle is \(180^{\circ}\). Let \(\angle M = 40^{\circ}\), \(\angle N=86^{\circ}\). Then \(\angle O=180^{\circ}-(40^{\circ} + 86^{\circ})=54^{\circ}\).

Step2: Relate angles and sides

In a triangle, the larger the angle, the longer the side opposite it.

• The side opposite \(\angle M\) is \(\overline{NO}\).
• The side opposite \(\angle N\) is \(\overline{OM}\).
• The side opposite \(\angle O\) is \(\overline{NM}\).

Since \(40^{\circ}<54^{\circ}<86^{\circ}\) (i.e., \(\angle M<\angle O<\angle N\)), then \(\overline{NO}<\overline{NM}<\overline{OM}\) (using the angle - side relationship in a triangle: if \(A,B,C\) are angles of a triangle and \(a,b,c\) are the sides opposite to \(A,B,C\) respectively, \(A < B < C\) implies \(a < b < c\)).

Answer:

E. \(\overline{NO},\overline{NM},\overline{OM}\)