Question

in triangle abc, points d and e are the midpoints of \\( \overline { a b } \\) and \\( \overline { a c } \\) respectively. what are the slope and length of \\( \overline { d e } \\), if the slope and length of \\( \overline { b c } \\) are 0.5 and 3.2 units, respectively?
a. slope = -2, length = 3.2 units
b. slope = 0.25, length = 3.2 units
c. slope = 0.5, length = 1.6 units
d. slope = -2, length = 1.6 units
e. slope = 0.25, length = 1.6 units

Explanation:

Step1: Recall the Midline Theorem

The Midline Theorem (also known as the Midsegment Theorem) states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long.

Step2: Determine the slope of \( \overline{DE} \)

Since \( \overline{DE} \) is parallel to \( \overline{BC} \) (by the Midline Theorem), parallel lines have the same slope. So the slope of \( \overline{DE} \) is equal to the slope of \( \overline{BC} \), which is \( 0.5 \).

Step3: Determine the length of \( \overline{DE} \)

By the Midline Theorem, the length of \( \overline{DE} \) is half the length of \( \overline{BC} \). Given the length of \( \overline{BC} \) is \( 3.2 \) units, the length of \( \overline{DE} \) is \( \frac{3.2}{2} = 1.6 \) units.

Answer:

C. slope = 0.5, length = 1.6 units