Question

name
1 - 3 reteach to build understanding
midpoint and distance

1. find the length of each segment in the path shown.

what is the total distance?
example: ( e(2,-1) ) and ( f(6,0) ). the distance ( d ) from point ( e ) to point ( f ) is
calculated with the distance formula.
\begin{aligned}d & =sqrt{left(x_2 - x_1
ight)^2+left(y_2 - y_1
ight)^2} \\& =sqrt{(6 - 2)^2+(0-(-1))^2} \\& =sqrt{(4)^2+(1)^2} \\& =sqrt{16 + 1} \\& =sqrt{17} approx 4.123end{aligned}

1. find the midpoint of each segment in exercise 1 above.

example: the midpoint ( m ) between point ( e ) and point ( f ) is calculated
with the midpoint formula.
m:left(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2}
ight)=left(\frac{6 + 2}{2},\frac{0+(-1)}{2}
ight)=left(\frac{8}{2},\frac{-1}{2}
ight)=left(4,\frac{-1}{2}
ight)
( overline{ab}:(quad) )
( overline{bc}:(quad) )
( overline{cd}:(quad) )
( overline{de}:(quad) )
( overline{ef}:(quad) )

1. cameron calculated the distance between point ( c ) and point ( d ) below.

is the distance correct? if not, what is her mistake?
\begin{aligned}d & =sqrt{(-2 + 2)^2+(6 + 3)^2} \\& =sqrt{(0)^2+(9)^2} \\& =9end{aligned}

Explanation:

1. Find the length of each segment and the total distance

• Coordinates of points:
• \(A(-6,6)\), \(B(-6,4)\), \(C(-2,4)\), \(D(0,1)\), \(E(2, - 1)\), \(F(6,0)\)
• Length of \(\overline{AB}\):
• Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(A(-6,6)\) and \(B(-6,4)\)
• \(x_1=-6,y_1 = 6,x_2=-6,y_2 = 4\)
• \(d=\sqrt{(-6+6)^2+(4 - 6)^2}=\sqrt{0+(-2)^2}=2\)
• Length of \(\overline{BC}\):
• For \(B(-6,4)\) and \(C(-2,4)\)
• \(x_1=-6,y_1 = 4,x_2=-2,y_2 = 4\)
• \(d=\sqrt{(-2 + 6)^2+(4 - 4)^2}=\sqrt{(4)^2+0}=4\)
• Length of \(\overline{CD}\):
• For \(C(-2,4)\) and \(D(0,1)\)
• \(x_1=-2,y_1 = 4,x_2=0,y_2 = 1\)
• \(d=\sqrt{(0 + 2)^2+(1 - 4)^2}=\sqrt{4+9}=\sqrt{13}\approx3.606\)
• Length of \(\overline{DE}\):
• For \(D(0,1)\) and \(E(2,-1)\)
• \(x_1=0,y_1 = 1,x_2=2,y_2=-1\)
• \(d=\sqrt{(2 - 0)^2+(-1 - 1)^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}\approx2.828\)
• Length of \(\overline{EF}\):
• Given \(E(2,-1)\) and \(F(6,0)\) (already calculated in the example as \(\sqrt{17}\approx4.123\))
• Total distance:
• \(d_{total}=2 + 4+\sqrt{13}+2\sqrt{2}+\sqrt{17}\approx2+4 + 3.606+2.828+4.123=16.557\)

2. Find the mid - point of each segment

• Mid - point formula: \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\)
• Mid - point of \(\overline{AB}\):
• For \(A(-6,6)\) and \(B(-6,4)\)
• \(M=(\frac{-6-6}{2},\frac{6 + 4}{2})=(-6,5)\)
• Mid - point of \(\overline{BC}\):
• For \(B(-6,4)\) and \(C(-2,4)\)
• \(M=(\frac{-6-2}{2},\frac{4 + 4}{2})=(-4,4)\)
• Mid - point of \(\overline{CD}\):
• For \(C(-2,4)\) and \(D(0,1)\)
• \(M=(\frac{-2+0}{2},\frac{4 + 1}{2})=(-1,\frac{5}{2})\)
• Mid - point of \(\overline{DE}\):
• For \(D(0,1)\) and \(E(2,-1)\)
• \(M=(\frac{0+2}{2},\frac{1-1}{2})=(1,0)\)
• Mid - point of \(\overline{EF}\):
• For \(E(2,-1)\) and \(F(6,0)\) (already calculated in the example as \((4,-\frac{1}{2})\))

3. Check Cameron's calculation

• Coordinates of \(C(-2,4)\) and \(D(0,1)\)
• Correct distance formula: \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), where \(x_1=-2,y_1 = 4,x_2=0,y_2 = 1\)
• Cameron's formula: \(d=\sqrt{(-2 + 2)^2+(6 + 3)^2}\), she used \(y_1 = 6\) instead of \(y_1 = 4\)
• Correct calculation:
• \(d=\sqrt{(0 + 2)^2+(1 - 4)^2}=\sqrt{4+9}=\sqrt{13}

eq9\)

Final Answers

1. Lengths: \(\overline{AB} = 2\), \(\overline{BC}=4\), \(\overline{CD}=\sqrt{13}\approx3.606\), \(\overline{DE}=2\sqrt{2}\approx2.828\), \(\overline{EF}=\sqrt{17}\approx4.123\); Total distance\(\approx16.557\)
2. Mid - points: \(\overline{AB}:(-6,5)\), \(\overline{BC}:(-4,4)\), \(\overline{CD}:(-1,\frac{5}{2})\), \(\overline{DE}:(1,0)\), \(\overline{EF}:(4,-\frac{1}{2})\)
3. Cameron's mistake: She used the wrong \(y\) - coordinate for point \(C\) (used \(y = 6\) instead of \(y = 4\)), and the correct distance between \(C\) and \(D\) is \(\sqrt{13}\)

Answer:

1. Find the length of each segment and the total distance

• Coordinates of points:
• \(A(-6,6)\), \(B(-6,4)\), \(C(-2,4)\), \(D(0,1)\), \(E(2, - 1)\), \(F(6,0)\)
• Length of \(\overline{AB}\):
• Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(A(-6,6)\) and \(B(-6,4)\)
• \(x_1=-6,y_1 = 6,x_2=-6,y_2 = 4\)
• \(d=\sqrt{(-6+6)^2+(4 - 6)^2}=\sqrt{0+(-2)^2}=2\)
• Length of \(\overline{BC}\):
• For \(B(-6,4)\) and \(C(-2,4)\)
• \(x_1=-6,y_1 = 4,x_2=-2,y_2 = 4\)
• \(d=\sqrt{(-2 + 6)^2+(4 - 4)^2}=\sqrt{(4)^2+0}=4\)
• Length of \(\overline{CD}\):
• For \(C(-2,4)\) and \(D(0,1)\)
• \(x_1=-2,y_1 = 4,x_2=0,y_2 = 1\)
• \(d=\sqrt{(0 + 2)^2+(1 - 4)^2}=\sqrt{4+9}=\sqrt{13}\approx3.606\)
• Length of \(\overline{DE}\):
• For \(D(0,1)\) and \(E(2,-1)\)
• \(x_1=0,y_1 = 1,x_2=2,y_2=-1\)
• \(d=\sqrt{(2 - 0)^2+(-1 - 1)^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}\approx2.828\)
• Length of \(\overline{EF}\):
• Given \(E(2,-1)\) and \(F(6,0)\) (already calculated in the example as \(\sqrt{17}\approx4.123\))
• Total distance:
• \(d_{total}=2 + 4+\sqrt{13}+2\sqrt{2}+\sqrt{17}\approx2+4 + 3.606+2.828+4.123=16.557\)

2. Find the mid - point of each segment

• Mid - point formula: \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\)
• Mid - point of \(\overline{AB}\):
• For \(A(-6,6)\) and \(B(-6,4)\)
• \(M=(\frac{-6-6}{2},\frac{6 + 4}{2})=(-6,5)\)
• Mid - point of \(\overline{BC}\):
• For \(B(-6,4)\) and \(C(-2,4)\)
• \(M=(\frac{-6-2}{2},\frac{4 + 4}{2})=(-4,4)\)
• Mid - point of \(\overline{CD}\):
• For \(C(-2,4)\) and \(D(0,1)\)
• \(M=(\frac{-2+0}{2},\frac{4 + 1}{2})=(-1,\frac{5}{2})\)
• Mid - point of \(\overline{DE}\):
• For \(D(0,1)\) and \(E(2,-1)\)
• \(M=(\frac{0+2}{2},\frac{1-1}{2})=(1,0)\)
• Mid - point of \(\overline{EF}\):
• For \(E(2,-1)\) and \(F(6,0)\) (already calculated in the example as \((4,-\frac{1}{2})\))

3. Check Cameron's calculation

• Coordinates of \(C(-2,4)\) and \(D(0,1)\)
• Correct distance formula: \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), where \(x_1=-2,y_1 = 4,x_2=0,y_2 = 1\)
• Cameron's formula: \(d=\sqrt{(-2 + 2)^2+(6 + 3)^2}\), she used \(y_1 = 6\) instead of \(y_1 = 4\)
• Correct calculation:
• \(d=\sqrt{(0 + 2)^2+(1 - 4)^2}=\sqrt{4+9}=\sqrt{13}

eq9\)

Final Answers

1. Lengths: \(\overline{AB} = 2\), \(\overline{BC}=4\), \(\overline{CD}=\sqrt{13}\approx3.606\), \(\overline{DE}=2\sqrt{2}\approx2.828\), \(\overline{EF}=\sqrt{17}\approx4.123\); Total distance\(\approx16.557\)
2. Mid - points: \(\overline{AB}:(-6,5)\), \(\overline{BC}:(-4,4)\), \(\overline{CD}:(-1,\frac{5}{2})\), \(\overline{DE}:(1,0)\), \(\overline{EF}:(4,-\frac{1}{2})\)
3. Cameron's mistake: She used the wrong \(y\) - coordinate for point \(C\) (used \(y = 6\) instead of \(y = 4\)), and the correct distance between \(C\) and \(D\) is \(\sqrt{13}\)