Question

1. find the midpoint of points a(-8, -4) and b(-1, 4) graphically: (a) plot the line segment $overline{ab}$. (b) draw a rectangle with the segment as a diagonal. (c) use the sides of the rectangle to help you plot the midpoint. (d) state the coordinates of the midpoint.

find the midpoint of points a(6, -4) and (-2, 2) graphically: (a) plot the line segment $overline{ab}$. (b) draw a rectangle with the segment as a diagonal. (c) use the sides of the rectangle to help you plot the midpoint. (d) state the coordinates of the midpoint.

1. find the midpoint of the segment with the following endpoints. (-2, -4) and (-6, 4)
2. find the midpoint of the segment with the following endpoints. (-1, 2) and (8, 6)
3. find the midpoint of the segment with the following endpoints. (3, -10) and (-2, -5)
4. find the midpoint of the segment with the following endpoints. (-8, 1) and (2, 7)

Explanation:

1. For points \(A(-8,-4)\) and \(B(-1,4)\):

• The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
• Step 1: Calculate the x - coordinate of the mid - point
• The x - coordinates of the points are \(x_1=-8\) and \(x_2 = - 1\).
• Using the formula \(x_m=\frac{x_1 + x_2}{2}\), we have \(x_m=\frac{-8+( - 1)}{2}=\frac{-8 - 1}{2}=\frac{-9}{2}=-4.5\).
• Step 2: Calculate the y - coordinate of the mid - point
• The y - coordinates of the points are \(y_1=-4\) and \(y_2 = 4\).
• Using the formula \(y_m=\frac{y_1 + y_2}{2}\), we have \(y_m=\frac{-4 + 4}{2}=0\).
• Answer: \((-4.5,0)\)

1. For points \((-2,-4)\) and \((-6,4)\):

• Step 1: Calculate the x - coordinate of the mid - point
• \(x_1=-2\) and \(x_2=-6\).
• \(x_m=\frac{-2+( - 6)}{2}=\frac{-2-6}{2}=\frac{-8}{2}=-4\).
• Step 2: Calculate the y - coordinate of the mid - point
• \(y_1=-4\) and \(y_2 = 4\).
• \(y_m=\frac{-4 + 4}{2}=0\).
• Answer: \((-4,0)\)

1. For points \((-1,2)\) and \((8,6)\):

• Step 1: Calculate the x - coordinate of the mid - point
• \(x_1=-1\) and \(x_2 = 8\).
• \(x_m=\frac{-1 + 8}{2}=\frac{7}{2}=3.5\).
• Step 2: Calculate the y - coordinate of the mid - point
• \(y_1=2\) and \(y_2 = 6\).
• \(y_m=\frac{2+6}{2}=\frac{8}{2}=4\).
• Answer: \((3.5,4)\)

1. For points \(A(6,-4)\) and \(B(-2,2)\):

• Step 1: Calculate the x - coordinate of the mid - point
• \(x_1 = 6\) and \(x_2=-2\).
• \(x_m=\frac{6+( - 2)}{2}=\frac{6 - 2}{2}=\frac{4}{2}=2\).
• Step 2: Calculate the y - coordinate of the mid - point
• \(y_1=-4\) and \(y_2 = 2\).
• \(y_m=\frac{-4 + 2}{2}=\frac{-2}{2}=-1\).
• Answer: \((2,-1)\)

1. For points \((3,-10)\) and \((-2,-5)\):

• Step 1: Calculate the x - coordinate of the mid - point
• \(x_1 = 3\) and \(x_2=-2\).
• \(x_m=\frac{3+( - 2)}{2}=\frac{3 - 2}{2}=\frac{1}{2}=0.5\).
• Step 2: Calculate the y - coordinate of the mid - point
• \(y_1=-10\) and \(y_2=-5\).
• \(y_m=\frac{-10+( - 5)}{2}=\frac{-10-5}{2}=\frac{-15}{2}=-7.5\).
• Answer: \((0.5,-7.5)\)

1. For points \((-8,1)\) and \((2,7)\):

• Step 1: Calculate the x - coordinate of the mid - point
• \(x_1=-8\) and \(x_2 = 2\).
• \(x_m=\frac{-8 + 2}{2}=\frac{-6}{2}=-3\).
• Step 2: Calculate the y - coordinate of the mid - point
• \(y_1=1\) and \(y_2 = 7\).
• \(y_m=\frac{1+7}{2}=\frac{8}{2}=4\).
• Answer: \((-3,4)\)

Answer:

1. For points \(A(-8,-4)\) and \(B(-1,4)\):

• The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
• Step 1: Calculate the x - coordinate of the mid - point
• The x - coordinates of the points are \(x_1=-8\) and \(x_2 = - 1\).
• Using the formula \(x_m=\frac{x_1 + x_2}{2}\), we have \(x_m=\frac{-8+( - 1)}{2}=\frac{-8 - 1}{2}=\frac{-9}{2}=-4.5\).
• Step 2: Calculate the y - coordinate of the mid - point
• The y - coordinates of the points are \(y_1=-4\) and \(y_2 = 4\).
• Using the formula \(y_m=\frac{y_1 + y_2}{2}\), we have \(y_m=\frac{-4 + 4}{2}=0\).
• Answer: \((-4.5,0)\)

1. For points \((-2,-4)\) and \((-6,4)\):

• Step 1: Calculate the x - coordinate of the mid - point
• \(x_1=-2\) and \(x_2=-6\).
• \(x_m=\frac{-2+( - 6)}{2}=\frac{-2-6}{2}=\frac{-8}{2}=-4\).
• Step 2: Calculate the y - coordinate of the mid - point
• \(y_1=-4\) and \(y_2 = 4\).
• \(y_m=\frac{-4 + 4}{2}=0\).
• Answer: \((-4,0)\)

1. For points \((-1,2)\) and \((8,6)\):

• Step 1: Calculate the x - coordinate of the mid - point
• \(x_1=-1\) and \(x_2 = 8\).
• \(x_m=\frac{-1 + 8}{2}=\frac{7}{2}=3.5\).
• Step 2: Calculate the y - coordinate of the mid - point
• \(y_1=2\) and \(y_2 = 6\).
• \(y_m=\frac{2+6}{2}=\frac{8}{2}=4\).
• Answer: \((3.5,4)\)

1. For points \(A(6,-4)\) and \(B(-2,2)\):

• Step 1: Calculate the x - coordinate of the mid - point
• \(x_1 = 6\) and \(x_2=-2\).
• \(x_m=\frac{6+( - 2)}{2}=\frac{6 - 2}{2}=\frac{4}{2}=2\).
• Step 2: Calculate the y - coordinate of the mid - point
• \(y_1=-4\) and \(y_2 = 2\).
• \(y_m=\frac{-4 + 2}{2}=\frac{-2}{2}=-1\).
• Answer: \((2,-1)\)

1. For points \((3,-10)\) and \((-2,-5)\):

• Step 1: Calculate the x - coordinate of the mid - point
• \(x_1 = 3\) and \(x_2=-2\).
• \(x_m=\frac{3+( - 2)}{2}=\frac{3 - 2}{2}=\frac{1}{2}=0.5\).
• Step 2: Calculate the y - coordinate of the mid - point
• \(y_1=-10\) and \(y_2=-5\).
• \(y_m=\frac{-10+( - 5)}{2}=\frac{-10-5}{2}=\frac{-15}{2}=-7.5\).
• Answer: \((0.5,-7.5)\)

1. For points \((-8,1)\) and \((2,7)\):

• Step 1: Calculate the x - coordinate of the mid - point
• \(x_1=-8\) and \(x_2 = 2\).
• \(x_m=\frac{-8 + 2}{2}=\frac{-6}{2}=-3\).
• Step 2: Calculate the y - coordinate of the mid - point
• \(y_1=1\) and \(y_2 = 7\).
• \(y_m=\frac{1+7}{2}=\frac{8}{2}=4\).
• Answer: \((-3,4)\)