Question

points s and t are midpoints of the sides of triangle fgh.
what is gf?
6 cm
16 cm
4 cm
8 cm

Explanation:

Step1: Identify Midsegment Theorem

Since S and T are midpoints, ST is a midsegment of triangle FGH. By the Midsegment Theorem, the midsegment is parallel to the third side and half its length. Wait, actually, looking at the segments: SG = 4 cm, and S is a midpoint, so GH = 8 cm? Wait, no, let's check the segments on GF. Wait, S is the midpoint of GH (since the tick marks show SG = SH? Wait, no, the tick marks on GH: SG is 4 cm, and S is a midpoint, so GH = 8 cm? Wait, no, the segment ST is 8 cm? Wait, no, the diagram: S is on GH, T is on FH. Wait, the Midsegment Theorem: In a triangle, the segment connecting the midpoints of two sides is parallel to the third side and half as long. Wait, here, S is the midpoint of GH (since SG = 4 cm, so GH = 8 cm? Wait, no, the problem: Points S and T are midpoints of the sides of triangle FGH. So S is the midpoint of GH, T is the midpoint of FH. Then ST is parallel to GF and ST = 1/2 GF? Wait, no, wait the labels: F, T, H on the horizontal line. Wait, FH: HT = 6 cm, and T is a midpoint, so TF = 6 cm, so FH = 12 cm? No, the tick marks on FH: HT and TF are equal, so HT = TF = 6 cm, so FH = 12 cm. Then S is the midpoint of GH: SG = 4 cm, so GH = 8 cm. Wait, but the segment ST: length 8 cm? Wait, no, the segment from S to T is 8 cm? Wait, maybe I got the sides wrong. Wait, the question is about GF. Wait, looking at the segment GF: S is a point on GH, and T is a point on FH, and ST is connected. Wait, maybe S is the midpoint of GH, T is the midpoint of FH, so ST is the midsegment, so ST is parallel to GF and ST = 1/2 GF. Wait, but ST is 8 cm? Then GF would be 16 cm? No, wait, maybe the other way. Wait, no, let's re-examine. Wait, the segment from G to S to H: SG = 4 cm, and S is the midpoint, so GH = 8 cm. Then ST is 8 cm? Wait, no, the problem is asking for GF. Wait, maybe the segment from G to F: S is on GH, T is on FH, and ST is 8 cm. Wait, if S and T are midpoints, then by the Midsegment Theorem, ST is parallel to GF and ST = 1/2 GF. Wait, but ST is 8 cm? Then GF would be 16 cm? Wait, but let's check the answer choices. The options are 6, 16, 4, 8. Wait, maybe I mixed up the midsegment. Wait, no, maybe S is the midpoint of GF? No, the labels: G, S, H; F, T, H. Wait, maybe the triangle is FGH, with base FH, and sides FG and GH. S is the midpoint of GH, T is the midpoint of FH. Then ST is the midsegment, so ST is parallel to FG and ST = 1/2 FG. Wait, ST is 8 cm, so FG (which is GF) would be 16 cm. Yes, that makes sense. So GF = 2 * ST. Since ST is 8 cm, GF = 16 cm.

Step2: Apply Midsegment Theorem

The Midsegment Theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. Here, S is the midpoint of \( \overline{GH} \) and T is the midpoint of \( \overline{FH} \), so \( \overline{ST} \) is the midsegment. Thus, \( ST = \frac{1}{2} GF \). Given \( ST = 8 \, \text{cm} \), we solve for \( GF \):
\( GF = 2 \times ST \)
\( GF = 2 \times 8 \, \text{cm} = 16 \, \text{cm} \).

Answer:

16 cm (corresponding to the option with "16 cm")