Question

a. name the midsegments and sides that are parallel in $delta abc$.
b. name the midsegments that are half as long as each side of the $delta abc$.

a. name the midsegments and sides that are parallel in $delta abc$. select all that apply.
$square$ a. $overline{bc}||overline{de}$
$square$ b. $overline{ab}||overline{df}$
$square$ c. $overline{bc}||overline{ef}$
$square$ d. $overline{ac}||overline{de}$
$square$ e. $overline{ac}||overline{df}$
$square$ f. $overline{ab}||overline{ef}$
$square$ g. there are no parallel segments

Explanation:

Step1: Identify midsegments via markings

From the diagram:

• $D$ is midpoint of $\overline{AB}$, $E$ is midpoint of $\overline{AC}$, $F$ is midpoint of $\overline{BC}$.
• Midsegments are $\overline{DE}$, $\overline{DF}$, $\overline{EF}$.

Step2: Apply Midsegment Theorem

The Midsegment Theorem states a triangle midsegment is parallel to the opposite side:

• $\overline{DE}$ (connects $AB, AC$ midpoints) $\parallel \overline{BC}$
• $\overline{DF}$ (connects $AB, BC$ midpoints) $\parallel \overline{AC}$
• $\overline{EF}$ (connects $AC, BC$ midpoints) $\parallel \overline{AB}$

Step3: Match to options

Parallel pairs: $\overline{BC}\parallel\overline{DE}$, $\overline{AC}\parallel\overline{DF}$, $\overline{AB}\parallel\overline{EF}$

For part b, by the Midsegment Theorem, each midsegment is half the length of the side it is parallel to:

• $\overline{DE}$ is half the length of $\overline{BC}$
• $\overline{DF}$ is half the length of $\overline{AC}$
• $\overline{EF}$ is half the length of $\overline{AB}$

Answer:

A. $\overline{BC}\parallel\overline{DE}$
E. $\overline{AC}\parallel\overline{DF}$
F. $\overline{AB}\parallel\overline{EF}$

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