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To find the slope of the segment connecting the midpoints of sides [tex]\( \overline{RS} \)[/tex] and [tex]\( \overline{RT} \)[/tex] in triangle [tex]\( \triangle RST \)[/tex], follow these steps:
1. Find the midpoint of side [tex]\( \overline{RS} \)[/tex].
- The coordinates of [tex]\( R \)[/tex] are [tex]\((-1, 10)\)[/tex].
- The coordinates of [tex]\( S \)[/tex] are [tex]\((5, 4)\)[/tex].
The formula for the midpoint [tex]\( M \)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Applying this formula to [tex]\( R \)[/tex] and [tex]\( S \)[/tex]:
[tex]\[ \text{Midpoint of } \overline{RS} = \left( \frac{-1 + 5}{2}, \frac{10 + 4}{2} \right) = \left( \frac{4}{2}, \frac{14}{2} \right) = (2.0, 7.0) \][/tex]
2. Find the midpoint of side [tex]\( \overline{RT} \)[/tex].
- The coordinates of [tex]\( R \)[/tex] are [tex]\((-1, 10)\)[/tex].
- The coordinates of [tex]\( T \)[/tex] are [tex]\((-4, -2)\)[/tex].
Using the midpoint formula:
[tex]\[ \text{Midpoint of } \overline{RT} = \left( \frac{-1 + -4}{2}, \frac{10 + -2}{2} \right) = \left( \frac{-5}{2}, \frac{8}{2} \right) = (-2.5, 4.0) \][/tex]
3. Calculate the slope of the segment connecting these midpoints.
- The coordinates of the midpoint of [tex]\( \overline{RS} \)[/tex] are [tex]\( (2.0, 7.0) \)[/tex].
- The coordinates of the midpoint of [tex]\( \overline{RT} \)[/tex] are [tex]\( (-2.5, 4.0) \)[/tex].
The slope [tex]\( m \)[/tex] of a line passing through points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates of the midpoints:
[tex]\[ m = \frac{4.0 - 7.0}{-2.5 - 2.0} = \frac{-3.0}{-4.5} = \frac{-3.0}{-4.5} = \frac{2}{3} \][/tex]
4. The slope of the segment connecting the midpoints of [tex]\( \overline{RS} \)[/tex] and [tex]\( \overline{RT} \)[/tex] is [tex]\( \frac{2}{3} \)[/tex].
Therefore, the answer is:
[tex]\[ \boxed{\frac{2}{3}} \][/tex]
1. Find the midpoint of side [tex]\( \overline{RS} \)[/tex].
- The coordinates of [tex]\( R \)[/tex] are [tex]\((-1, 10)\)[/tex].
- The coordinates of [tex]\( S \)[/tex] are [tex]\((5, 4)\)[/tex].
The formula for the midpoint [tex]\( M \)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Applying this formula to [tex]\( R \)[/tex] and [tex]\( S \)[/tex]:
[tex]\[ \text{Midpoint of } \overline{RS} = \left( \frac{-1 + 5}{2}, \frac{10 + 4}{2} \right) = \left( \frac{4}{2}, \frac{14}{2} \right) = (2.0, 7.0) \][/tex]
2. Find the midpoint of side [tex]\( \overline{RT} \)[/tex].
- The coordinates of [tex]\( R \)[/tex] are [tex]\((-1, 10)\)[/tex].
- The coordinates of [tex]\( T \)[/tex] are [tex]\((-4, -2)\)[/tex].
Using the midpoint formula:
[tex]\[ \text{Midpoint of } \overline{RT} = \left( \frac{-1 + -4}{2}, \frac{10 + -2}{2} \right) = \left( \frac{-5}{2}, \frac{8}{2} \right) = (-2.5, 4.0) \][/tex]
3. Calculate the slope of the segment connecting these midpoints.
- The coordinates of the midpoint of [tex]\( \overline{RS} \)[/tex] are [tex]\( (2.0, 7.0) \)[/tex].
- The coordinates of the midpoint of [tex]\( \overline{RT} \)[/tex] are [tex]\( (-2.5, 4.0) \)[/tex].
The slope [tex]\( m \)[/tex] of a line passing through points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates of the midpoints:
[tex]\[ m = \frac{4.0 - 7.0}{-2.5 - 2.0} = \frac{-3.0}{-4.5} = \frac{-3.0}{-4.5} = \frac{2}{3} \][/tex]
4. The slope of the segment connecting the midpoints of [tex]\( \overline{RS} \)[/tex] and [tex]\( \overline{RT} \)[/tex] is [tex]\( \frac{2}{3} \)[/tex].
Therefore, the answer is:
[tex]\[ \boxed{\frac{2}{3}} \][/tex]
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