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1. find the midpoint of 2. find the distance between 3. given the endpo…

Question

  1. find the midpoint of
  2. find the distance between
  3. given the endpoint (-2/3, -1 1/2) and the midpoint (-1 1/8, 1 1/3). find the other endpoint.
  4. find the midpoint of: (-1,3) & (1,-2)
  5. find the distance between: (-6.1,-4.8) & (5.1,-0.6)
  6. find the distance between: (10,1) & (9,-4)
  7. find the midpoint of: (-4,4) & (-2,2)
  8. given the endpoint (5,2) and the midpoint (-10,-2). find the other endpoint.
  9. find the distance between: (5.7,-4.1) & (8,5.3)
  10. find the midpoint of: (4,7) & (-2,1)
  11. find the distance between: (-3,-1) & (1,-1)
  12. find the distance between: (0,-1) & (2,0)

light green 9.7
dark green √26
(-1 7/12, 4 5/6) light blue (-25,-6)
light blue (1.05,4.25)
dark blue (0,0.5)
purple 2√10
light brown (-3)
dark brown (1,4)
white 12

Explanation:

Response
  1. Mid - point formula: The mid - point \(M\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). The distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
  • Problem 1:
  • Let the two points be \((x_1,y_1)\) and \((x_2,y_2)\). The mid - point formula is \(M = (\frac{x_1+x_2}{2},\frac{y_1 + y_2}{2})\). But the points are not given clearly in the problem statement.
  • Problem 2:
  • Let \((x_1,y_1)\) and \((x_2,y_2)\) be the two points. The distance \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Since the points are not clear, we can't calculate.
  • Problem 3:
  • Let one endpoint be \((x_1,y_1)=(-\frac{2}{3},-1\frac{1}{2})=(-\frac{2}{3},-\frac{3}{2})\) and the mid - point \((x_m,y_m)=(-1\frac{1}{8},1\frac{1}{3})=(-\frac{9}{8},\frac{4}{3})\).
  • Step 1: Use the mid - point formula for \(x\) - coordinate
  • We know that \(x_m=\frac{x_1 + x_2}{2}\), so \(-\frac{9}{8}=\frac{-\frac{2}{3}+x_2}{2}\). Cross - multiply: \(-\frac{9}{8}\times2=-\frac{2}{3}+x_2\), \(-\frac{9}{4}=-\frac{2}{3}+x_2\). Then \(x_2=-\frac{9}{4}+\frac{2}{3}=\frac{-27 + 8}{12}=-\frac{19}{12}=-1\frac{7}{12}\).
  • Step 2: Use the mid - point formula for \(y\) - coordinate
  • We know that \(y_m=\frac{y_1 + y_2}{2}\), so \(\frac{4}{3}=\frac{-\frac{3}{2}+y_2}{2}\). Cross - multiply: \(\frac{4}{3}\times2=-\frac{3}{2}+y_2\), \(\frac{8}{3}=-\frac{3}{2}+y_2\). Then \(y_2=\frac{8}{3}+\frac{3}{2}=\frac{16 + 9}{6}=\frac{25}{6}=4\frac{1}{6}\). The other endpoint is \((-\frac{19}{12},\frac{25}{6})\).
  • Problem 4:
  • Let \((x_1,y_1)=(-1,3)\) and \((x_2,y_2)=(1,-2)\).
  • Step 1: Calculate the \(x\) - coordinate of the mid - point
  • \(x_m=\frac{x_1 + x_2}{2}=\frac{-1 + 1}{2}=0\).
  • Step 2: Calculate the \(y\) - coordinate of the mid - point
  • \(y_m=\frac{y_1 + y_2}{2}=\frac{3+( - 2)}{2}=\frac{1}{2}\). The mid - point is \((0,\frac{1}{2})\).
  • Problem 5:
  • Let \((x_1,y_1)=(-6.1,-4.8)\) and \((x_2,y_2)=(5.1,-0.6)\).
  • Step 1: Calculate \((x_2 - x_1)\) and \((y_2 - y_1)\)
  • \(x_2 - x_1=5.1-( - 6.1)=11.2\), \(y_2 - y_1=-0.6-( - 4.8)=4.2\).
  • Step 2: Use the distance formula
  • \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=\sqrt{(11.2)^2+(4.2)^2}=\sqrt{125.44 + 17.64}=\sqrt{143.08}\approx11.96\).
  • Problem 6:
  • Let \((x_1,y_1)=(10,1)\) and \((x_2,y_2)=(9,-4)\).
  • Step 1: Calculate \((x_2 - x_1)\) and \((y_2 - y_1)\)
  • \(x_2 - x_1=9 - 10=-1\), \(y_2 - y_1=-4 - 1=-5\).
  • Step 2: Use the distance formula
  • \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=\sqrt{(-1)^2+( - 5)^2}=\sqrt{1 + 25}=\sqrt{26}\).
  • Problem 7:
  • Let \((x_1,y_1)=(-4,4)\) and \((x_2,y_2)=(-2,2)\).
  • Step 1: Calculate the \(x\) - coordinate of the mid - point
  • \(x_m=\frac{x_1 + x_2}{2}=\frac{-4+( - 2)}{2}=\frac{-6}{2}=-3\).
  • Step 2: Calculate the \(y\) - coordinate of the mid - point
  • \(y_m=\frac{y_1 + y_2}{2}=\frac{4 + 2}{2}=3\). The mid - point is \((-3,3)\).
  • Problem 8:
  • Let one endpoint \((x_1,y_1)=(5,2)\) and the mid - point \((x_m,y_m)=(-10,-2)\).
  • Step 1: Use the mid - point formula for \(x\) - coordinate
  • \(x_m=\frac{x_1 + x_2}{2}\), so \(-10=\frac{5 + x_2}{2}\). Cross - multiply: \(-10\times2=5 + x_2\), \(x_2=-20 - 5=-25\).
  • Step 2: Use the mid - point formula for \(y\) - coordinate
  • \(…

Answer:

  1. No clear answer as points not given.
  2. No clear answer as points not given.
  3. \((-\frac{19}{12},\frac{25}{6})\)
  4. \((0,\frac{1}{2})\)
  5. \(\sqrt{143.08}\approx11.96\)
  6. \(\sqrt{26}\)
  7. \((-3,3)\)
  8. \((-25,-6)\)
  9. \(\sqrt{93.65}\approx9.68\)
  10. \((1,4)\)
  11. \(4\)
  12. \(\sqrt{5}\)