Question

1. determine the mid - point coordinates and the length of each line segment.

Explanation:

1. Mid - point formula:

• The mid - point formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\), and the distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

1. For the line segment with endpoints \(G(-2,4)\) and \(H(6,4)\):

• Mid - point:
• ## Step 1: Calculate the x - coordinate of the mid - point
• Using the mid - point formula \(x_m=\frac{-2 + 6}{2}=\frac{4}{2}=2\).
• ## Step 2: Calculate the y - coordinate of the mid - point
• \(y_m=\frac{4 + 4}{2}=\frac{8}{2}=4\). So the mid - point is \((2,4)\).
• Length:
• ## Step 3: Calculate the length using the distance formula
• \(d=\sqrt{(6-(-2))^2+(4 - 4)^2}=\sqrt{(6 + 2)^2+0^2}=\sqrt{8^2}=8\).

1. For the line segment with endpoints \(I(-2,-5)\) and \(J(3,1)\):

• Mid - point:
• ## Step 4: Calculate the x - coordinate of the mid - point
• \(x_m=\frac{-2+3}{2}=\frac{1}{2}\).
• ## Step 5: Calculate the y - coordinate of the mid - point
• \(y_m=\frac{-5 + 1}{2}=\frac{-4}{2}=-2\). So the mid - point is \((\frac{1}{2},-2)\).
• Length:
• ## Step 6: Calculate the length using the distance formula
• \(d=\sqrt{(3-(-2))^2+(1-(-5))^2}=\sqrt{(3 + 2)^2+(1 + 5)^2}=\sqrt{5^2+6^2}=\sqrt{25 + 36}=\sqrt{61}\).

1. For the line segment with endpoints \(K(4,-2)\) and \(L(6,-5)\):

• Mid - point:
• ## Step 7: Calculate the x - coordinate of the mid - point
• \(x_m=\frac{4 + 6}{2}=\frac{10}{2}=5\).
• ## Step 8: Calculate the y - coordinate of the mid - point
• \(y_m=\frac{-2+(-5)}{2}=\frac{-2-5}{2}=-\frac{7}{2}\). So the mid - point is \((5,-\frac{7}{2})\).
• Length:
• ## Step 9: Calculate the length using the distance formula
• \(d=\sqrt{(6 - 4)^2+(-5-(-2))^2}=\sqrt{2^2+(-5 + 2)^2}=\sqrt{4+(-3)^2}=\sqrt{4 + 9}=\sqrt{13}\).

1. For the line segment with endpoints \(F(-3,3)\) and \(E(-3,-4)\):

• Mid - point:
• ## Step 10: Calculate the x - coordinate of the mid - point
• \(x_m=\frac{-3+(-3)}{2}=\frac{-6}{2}=-3\).
• ## Step 11: Calculate the y - coordinate of the mid - point
• \(y_m=\frac{3+(-4)}{2}=\frac{3 - 4}{2}=-\frac{1}{2}\). So the mid - point is \((-3,-\frac{1}{2})\).
• Length:
• ## Step 12: Calculate the length using the distance formula
• \(d=\sqrt{(-3-(-3))^2+(-4 - 3)^2}=\sqrt{0^2+(-7)^2}=\sqrt{49}=7\).

Answer:

For \(G(-2,4)\) and \(H(6,4)\): Mid - point \((2,4)\), Length \(8\);
For \(I(-2,-5)\) and \(J(3,1)\): Mid - point \((\frac{1}{2},-2)\), Length \(\sqrt{61}\);
For \(K(4,-2)\) and \(L(6,-5)\): Mid - point \((5,-\frac{7}{2})\), Length \(\sqrt{13}\);
For \(F(-3,3)\) and \(E(-3,-4)\): Mid - point \((-3,-\frac{1}{2})\), Length \(7\)