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Sagot :
Let's find the midpoint of each line segment with the specified endpoints. The formula to find 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]\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Let's apply this formula to each pair of endpoints:
9) Find the midpoint of the line segment with endpoints [tex]\((-4, 4)\)[/tex] and [tex]\((5, -1)\)[/tex]:
[tex]\[ \left( \frac{-4 + 5}{2}, \frac{4 + (-1)}{2} \right) = \left( \frac{1}{2}, \frac{3}{2} \right) = (0.5, 1.5) \][/tex]
10) Find the midpoint of the line segment with endpoints [tex]\((-1, -6)\)[/tex] and [tex]\((-6, 5)\)[/tex]:
[tex]\[ \left( \frac{-1 + (-6)}{2}, \frac{-6 + 5}{2} \right) = \left( \frac{-7}{2}, \frac{-1}{2} \right) = (-3.5, -0.5) \][/tex]
11) Find the midpoint of the line segment with endpoints [tex]\((2, 4)\)[/tex] and [tex]\((1, -3)\)[/tex]:
[tex]\[ \left( \frac{2 + 1}{2}, \frac{4 + (-3)}{2} \right) = \left( \frac{3}{2}, \frac{1}{2} \right) = (1.5, 0.5) \][/tex]
12) Find the midpoint of the line segment with endpoints [tex]\((-4, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex]:
[tex]\[ \left( \frac{-4 + (-2)}{2}, \frac{4 + 2}{2} \right) = \left( \frac{-6}{2}, \frac{6}{2} \right) = (-3, 3) \][/tex]
13) Find the midpoint of the line segment with endpoints [tex]\((5, 2)\)[/tex] and [tex]\((-4, -3)\)[/tex]:
[tex]\[ \left( \frac{5 + (-4)}{2}, \frac{2 + (-3)}{2} \right) = \left( \frac{1}{2}, \frac{-1}{2} \right) = (0.5, -0.5) \][/tex]
14) Find the midpoint of the line segment with endpoints [tex]\((-1, 1)\)[/tex] and [tex]\((5, -5)\)[/tex]:
[tex]\[ \left( \frac{-1 + 5}{2}, \frac{1 + (-5)}{2} \right) = \left( \frac{4}{2}, \frac{-4}{2} \right) = (2, -2) \][/tex]
15) Find the midpoint of the line segment with endpoints [tex]\((2, -1)\)[/tex] and [tex]\((-6, 0)\)[/tex]:
[tex]\[ \left( \frac{2 + (-6)}{2}, \frac{-1 + 0}{2} \right) = \left( \frac{-4}{2}, \frac{-1}{2} \right) = (-2, -0.5) \][/tex]
16) Find the midpoint of the line segment with endpoints [tex]\((-3.1, -2.8)\)[/tex] and [tex]\((-4.92, -3.3)\)[/tex]:
[tex]\[ \left( \frac{-3.1 + (-4.92)}{2}, \frac{-2.8 + (-3.3)}{2} \right) = \left( \frac{-8.02}{2}, \frac{-6.1}{2} \right) = (-4.01, -3.05) \][/tex]
17) Find the midpoint of the line segment with endpoints [tex]\((-5.1, -2)\)[/tex] and [tex]\((1.4, 1.7)\)[/tex]:
[tex]\[ \left( \frac{-5.1 + 1.4}{2}, \frac{-2 + 1.7}{2} \right) = \left( \frac{-3.7}{2}, \frac{-0.3}{2} \right) = (-1.85, -0.15) \][/tex]
18) Find the midpoint of the line segment with endpoints [tex]\((4.9, -1.3)\)[/tex] and [tex]\((-5.2, -0.6)\)[/tex]:
[tex]\[ \left( \frac{4.9 + (-5.2)}{2}, \frac{-1.3 + (-0.6)}{2} \right) = \left( \frac{-0.3}{2}, \frac{-1.9}{2} \right) = (-0.15, -0.95) \][/tex]
19) Find the midpoint of the line segment with endpoints [tex]\((5.1, 5.71)\)[/tex] and [tex]\((6, 3.6)\)[/tex]:
[tex]\[ \left( \frac{5.1 + 6}{2}, \frac{5.71 + 3.6}{2} \right) = \left( \frac{11.1}{2}, \frac{9.31}{2} \right) = (5.55, 4.655) \][/tex]
20) Find the midpoint of the line segment with endpoints [tex]\((3.1, -2.1)\)[/tex] and [tex]\((-0.52, -0.6)\)[/tex]:
[tex]\[ \left( \frac{3.1 + (-0.52)}{2}, \frac{-2.1 + (-0.6)}{2} \right) = \left( \frac{2.58}{2}, \frac{-2.7}{2} \right) = (1.29, -1.35) \][/tex]
So the midpoints for each pair of endpoints are:
1. [tex]\((0.5, 1.5)\)[/tex]
2. [tex]\((-3.5, -0.5)\)[/tex]
3. [tex]\((1.5, 0.5)\)[/tex]
4. [tex]\((-3.0, 3.0)\)[/tex]
5. [tex]\((0.5, -0.5)\)[/tex]
6. [tex]\((2.0, -2.0)\)[/tex]
7. [tex]\((-2.0, -0.5)\)[/tex]
8. [tex]\((-4.01, -3.05)\)[/tex]
9. [tex]\((-1.85, -0.15)\)[/tex]
10. [tex]\((-0.15, -0.95)\)[/tex]
11. [tex]\((5.55, 4.655)\)[/tex]
12. [tex]\((1.29, -1.35)\)[/tex]
[tex]\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Let's apply this formula to each pair of endpoints:
9) Find the midpoint of the line segment with endpoints [tex]\((-4, 4)\)[/tex] and [tex]\((5, -1)\)[/tex]:
[tex]\[ \left( \frac{-4 + 5}{2}, \frac{4 + (-1)}{2} \right) = \left( \frac{1}{2}, \frac{3}{2} \right) = (0.5, 1.5) \][/tex]
10) Find the midpoint of the line segment with endpoints [tex]\((-1, -6)\)[/tex] and [tex]\((-6, 5)\)[/tex]:
[tex]\[ \left( \frac{-1 + (-6)}{2}, \frac{-6 + 5}{2} \right) = \left( \frac{-7}{2}, \frac{-1}{2} \right) = (-3.5, -0.5) \][/tex]
11) Find the midpoint of the line segment with endpoints [tex]\((2, 4)\)[/tex] and [tex]\((1, -3)\)[/tex]:
[tex]\[ \left( \frac{2 + 1}{2}, \frac{4 + (-3)}{2} \right) = \left( \frac{3}{2}, \frac{1}{2} \right) = (1.5, 0.5) \][/tex]
12) Find the midpoint of the line segment with endpoints [tex]\((-4, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex]:
[tex]\[ \left( \frac{-4 + (-2)}{2}, \frac{4 + 2}{2} \right) = \left( \frac{-6}{2}, \frac{6}{2} \right) = (-3, 3) \][/tex]
13) Find the midpoint of the line segment with endpoints [tex]\((5, 2)\)[/tex] and [tex]\((-4, -3)\)[/tex]:
[tex]\[ \left( \frac{5 + (-4)}{2}, \frac{2 + (-3)}{2} \right) = \left( \frac{1}{2}, \frac{-1}{2} \right) = (0.5, -0.5) \][/tex]
14) Find the midpoint of the line segment with endpoints [tex]\((-1, 1)\)[/tex] and [tex]\((5, -5)\)[/tex]:
[tex]\[ \left( \frac{-1 + 5}{2}, \frac{1 + (-5)}{2} \right) = \left( \frac{4}{2}, \frac{-4}{2} \right) = (2, -2) \][/tex]
15) Find the midpoint of the line segment with endpoints [tex]\((2, -1)\)[/tex] and [tex]\((-6, 0)\)[/tex]:
[tex]\[ \left( \frac{2 + (-6)}{2}, \frac{-1 + 0}{2} \right) = \left( \frac{-4}{2}, \frac{-1}{2} \right) = (-2, -0.5) \][/tex]
16) Find the midpoint of the line segment with endpoints [tex]\((-3.1, -2.8)\)[/tex] and [tex]\((-4.92, -3.3)\)[/tex]:
[tex]\[ \left( \frac{-3.1 + (-4.92)}{2}, \frac{-2.8 + (-3.3)}{2} \right) = \left( \frac{-8.02}{2}, \frac{-6.1}{2} \right) = (-4.01, -3.05) \][/tex]
17) Find the midpoint of the line segment with endpoints [tex]\((-5.1, -2)\)[/tex] and [tex]\((1.4, 1.7)\)[/tex]:
[tex]\[ \left( \frac{-5.1 + 1.4}{2}, \frac{-2 + 1.7}{2} \right) = \left( \frac{-3.7}{2}, \frac{-0.3}{2} \right) = (-1.85, -0.15) \][/tex]
18) Find the midpoint of the line segment with endpoints [tex]\((4.9, -1.3)\)[/tex] and [tex]\((-5.2, -0.6)\)[/tex]:
[tex]\[ \left( \frac{4.9 + (-5.2)}{2}, \frac{-1.3 + (-0.6)}{2} \right) = \left( \frac{-0.3}{2}, \frac{-1.9}{2} \right) = (-0.15, -0.95) \][/tex]
19) Find the midpoint of the line segment with endpoints [tex]\((5.1, 5.71)\)[/tex] and [tex]\((6, 3.6)\)[/tex]:
[tex]\[ \left( \frac{5.1 + 6}{2}, \frac{5.71 + 3.6}{2} \right) = \left( \frac{11.1}{2}, \frac{9.31}{2} \right) = (5.55, 4.655) \][/tex]
20) Find the midpoint of the line segment with endpoints [tex]\((3.1, -2.1)\)[/tex] and [tex]\((-0.52, -0.6)\)[/tex]:
[tex]\[ \left( \frac{3.1 + (-0.52)}{2}, \frac{-2.1 + (-0.6)}{2} \right) = \left( \frac{2.58}{2}, \frac{-2.7}{2} \right) = (1.29, -1.35) \][/tex]
So the midpoints for each pair of endpoints are:
1. [tex]\((0.5, 1.5)\)[/tex]
2. [tex]\((-3.5, -0.5)\)[/tex]
3. [tex]\((1.5, 0.5)\)[/tex]
4. [tex]\((-3.0, 3.0)\)[/tex]
5. [tex]\((0.5, -0.5)\)[/tex]
6. [tex]\((2.0, -2.0)\)[/tex]
7. [tex]\((-2.0, -0.5)\)[/tex]
8. [tex]\((-4.01, -3.05)\)[/tex]
9. [tex]\((-1.85, -0.15)\)[/tex]
10. [tex]\((-0.15, -0.95)\)[/tex]
11. [tex]\((5.55, 4.655)\)[/tex]
12. [tex]\((1.29, -1.35)\)[/tex]
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