Question

geometry sem a- a
triangle similarity theorems
points o and n are midpoints of the sides of triangle def.
what is dm?
76 cm
30 cm
38 cm
22 cm

Explanation:

Step1: Identify Midsegment Theorem

In triangle \( DEF \), \( O \) and \( N \) are midpoints (given by mid - segment markings on \( DE \) and \( EF \)). By the Midsegment Theorem, the segment connecting midpoints of two sides of a triangle is parallel to the third side and half its length. Also, \( M \) should be the midpoint of \( DF \) because of the symmetry and the mid - segment properties.
Looking at the length of \( ON = 38\space cm \), and by the Midsegment Theorem, \( ON\parallel DF \) and \( ON=\frac{1}{2}DF \). But also, since \( O \) and \( N \) are midpoints, and the markings on \( DF \) suggest that \( DM = ON \) (because \( M \) is the midpoint and the segments are related by the mid - segment and the congruency of the divided parts of \( DF \)).

Step2: Determine the length of \( DM \)

From the diagram, \( ON = 38\space cm \). Since \( M \) is the midpoint of \( DF \) and \( O \) and \( N \) are midpoints of \( DE \) and \( EF \) respectively, the segment \( DM \) should be equal in length to \( ON \) (due to the properties of triangle midsegments and the congruency of the sub - segments of \( DF \) as indicated by the tick marks). So \( DM = 38\space cm \).

Answer:

38 cm