Question

a. name the midsegments and sides that are parallel in \\( \triangle a b c \\).
b. name the midsegments that are half as long as each side of the \\( \triangle a b c \\).
a. name the midsegments and sides that are parallel in \\( \triangle a b c \\). select all that apply.
a. \\( \overline{a b} \\| \overline{e f} \\) b. \\( \overline{a c} \\| \overline{d f} \\)
c. \\( \overline{b c} \\| \overline{e f} \\) d. \\( \overline{a c} \\| \overline{d e} \\)
e. \\( \overline{a b} \\| \overline{d f} \\) f. \\( \overline{b c} \\| \overline{d e} \\)
g. there are no parallel segments.
b. name the midsegments that are half as long as each side of the \\( \triangle a b c \\). select all that apply.
a. \\( a d=\frac{1}{2} a b \\) b. \\( d f=\frac{1}{2} a c \\)
c. \\( e f=\frac{1}{2} a b \\) d. \\( d e=\frac{1}{2} b c \\)
e. \\( f c=\frac{1}{2} b c \\) f. there are no midsegments that are half as long as any side of the \\( \triangle a b c \\).

Explanation:

Step1: Recall the mid - segment theorem

The mid - segment theorem states that a mid - segment of a triangle is parallel to a side of the triangle and its length is half of the length of that side. A mid - segment of a triangle is a segment that connects the midpoints of two sides of the triangle.

Step2: Identify mid - segments and parallel sides for part (a)

• For $\overline{EF}$: Since $E$ is the midpoint of $AC$ and $F$ is the midpoint of $BC$, by the mid - segment theorem, $\overline{EF}\parallel\overline{AB}$.
• For $\overline{DF}$: Since $D$ is the midpoint of $AB$ and $F$ is the midpoint of $BC$, by the mid - segment theorem, $\overline{DF}\parallel\overline{AC}$.
• For $\overline{DE}$: Since $D$ is the midpoint of $AB$ and $E$ is the midpoint of $AC$, by the mid - segment theorem, $\overline{DE}\parallel\overline{BC}$.

Step3: Identify mid - segments and their lengths for part (b)

• For $\overline{EF}$: Using the mid - segment theorem, if $\overline{EF}$ is a mid - segment (connects midpoints of $AC$ and $BC$), then $EF=\frac{1}{2}AB$.
• For $\overline{DF}$: Since $\overline{DF}$ is a mid - segment (connects midpoints of $AB$ and $BC$), then $DF = \frac{1}{2}AC$.
• For $\overline{DE}$: Since $\overline{DE}$ is a mid - segment (connects midpoints of $AB$ and $AC$), then $DE=\frac{1}{2}BC$.

Answer:

a. A. $\overline{AB}\parallel\overline{EF}$, B. $\overline{AC}\parallel\overline{DF}$, F. $\overline{BC}\parallel\overline{DE}$
b. B. $DF = \frac{1}{2}AC$, C. $EF=\frac{1}{2}AB$, D. $DE=\frac{1}{2}BC$