Question

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

Since O and N are midpoints, ON is the midsegment of triangle DEF. The Midsegment Theorem states that the midsegment is parallel to the third side (DF) and half its length. Wait, no—wait, also, looking at the markings: O is the midpoint of DE (since DE is marked with two equal segments, 22 cm each? Wait, no, DE has O as a midpoint, so DO = OE = 22 cm. N is the midpoint of EF? Wait, EF is 30 cm? No, wait, the segment ON is 38 cm. Wait, also, DF has M as a midpoint? Wait, the markings on DF: DM and MF have the same triple marks, so M is the midpoint of DF. Also, O is midpoint of DE, N is midpoint of EF, so ON is midsegment, so ON should be parallel to DF and ON = ½ DF. Wait, but also, looking at the sides: DE is split by O into two 22 cm segments, so DE = 44 cm? Wait, no, the diagram shows O on DE with DE split into two equal parts (single marks), so DO = OE = 22 cm, so DE = 44 cm. EF is split by N into two 30 cm segments (double marks), so EN = NF = 30 cm, so EF = 60 cm. Then ON is 38 cm. Now, DF: M is the midpoint (triple marks), so DM = MF. Also, since O and N are midpoints, ON is midsegment, so ON = ½ DF. Wait, but ON is 38 cm, so DF = 2 38 = 76 cm? No, that can't be. Wait, maybe I got the midsegment wrong. Wait, alternatively, maybe triangle ODN or something? Wait, no, the question is about DM. Wait, looking at the markings on DF: DM and MF have the same triple tick marks, so M is the midpoint. Also, O is midpoint of DE, N is midpoint of EF, so ON is midsegment, so ON is parallel to DF and ON = ½ DF. Wait, ON is 38 cm, so DF = 76 cm? But M is the midpoint, so DM = ½ DF = 38 cm? Wait, but let's check the other sides. Wait, DE: O is midpoint, so DO = 22 cm, so DE = 44 cm. EF: N is midpoint, EN = 30 cm, so EF = 60 cm. Then ON is midsegment, so ON should be ½ DF. So ON = 38 cm, so DF = 76 cm. Then M is midpoint of DF, so DM = 76 / 2 = 38 cm. Alternatively, maybe the midsegment is ON, so DF = 2 ON = 76, so DM (midpoint) is 38. So DM is 38 cm.

Step2: Confirm with Midpoint Markings

The segment DF has M as the midpoint (triple tick marks on DM and MF), so DM = MF. Also, ON is midsegment (O and N midpoints of DE and EF), so ON = ½ DF. Given ON = 38 cm, DF = 2 * 38 = 76 cm. Then DM = 76 / 2 = 38 cm.

Answer:

38 cm