Question

section 1.3
i can find a mid - point given a segment or given 2 points.
the endpoints of segments a and b are given. find the midpoint m.

1. a(3,9) and b(5,4)

find the missing endpoint given point a and mid - point m.

1. a(2,5) and m(3,4)

i can find the distance between two given points.
find the distance between the given points.
a(3, - 1) and b(7,9)

Explanation:

1. Find the mid - point of segment with endpoints \(A(3,9)\) and \(B(9,4)\):

• Answer: \((6,\frac{13}{2})\)
• Explanation:
• Step 1: Recall the mid - point formula
• The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Here, \(x_1 = 3,y_1=9,x_2 = 9,y_2 = 4\).
• Step 2: Calculate the x - coordinate of the mid - point
• \(x=\frac{3 + 9}{2}=\frac{12}{2}=6\).
• Step 3: Calculate the y - coordinate of the mid - point
• \(y=\frac{9+4}{2}=\frac{13}{2}\).

1. Find the missing endpoint given \(A(2,5)\) and mid - point \(M(3,4)\):

• Answer: \((4,3)\)
• Explanation:
• Step 1: Let the missing endpoint be \((x,y)\)
• Using the mid - point formula \(M=(\frac{x_1 + x}{2},\frac{y_1 + y}{2})\), where \((x_1,y_1)=(2,5)\) and \(M=(3,4)\).
• Step 2: Solve for \(x\)
• We have \(\frac{2 + x}{2}=3\). Cross - multiply: \(2 + x=6\), then \(x = 4\).
• Step 3: Solve for \(y\)
• We have \(\frac{5 + y}{2}=4\). Cross - multiply: \(5 + y=8\), then \(y = 3\).

1. Find the distance between \(A(3,-1)\) and \(B(7,9)\):

• Answer: \(2\sqrt{29}\)
• Explanation:
• Step 1: Recall the distance formula
• The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \(x_1 = 3,y_1=-1,x_2 = 7,y_2 = 9\).
• Step 2: Calculate \((x_2 - x_1)^2+(y_2 - y_1)^2\)
• \(x_2 - x_1=7 - 3 = 4\), \(y_2 - y_1=9-( - 1)=10\). Then \((x_2 - x_1)^2+(y_2 - y_1)^2=4^2+10^2=16 + 100 = 116\).
• Step 3: Calculate the distance \(d\)
• \(d=\sqrt{116}=\sqrt{4\times29}=2\sqrt{29}\).

Answer:

1. Find the mid - point of segment with endpoints \(A(3,9)\) and \(B(9,4)\):

• Answer: \((6,\frac{13}{2})\)
• Explanation:
• Step 1: Recall the mid - point formula
• The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Here, \(x_1 = 3,y_1=9,x_2 = 9,y_2 = 4\).
• Step 2: Calculate the x - coordinate of the mid - point
• \(x=\frac{3 + 9}{2}=\frac{12}{2}=6\).
• Step 3: Calculate the y - coordinate of the mid - point
• \(y=\frac{9+4}{2}=\frac{13}{2}\).

1. Find the missing endpoint given \(A(2,5)\) and mid - point \(M(3,4)\):

• Answer: \((4,3)\)
• Explanation:
• Step 1: Let the missing endpoint be \((x,y)\)
• Using the mid - point formula \(M=(\frac{x_1 + x}{2},\frac{y_1 + y}{2})\), where \((x_1,y_1)=(2,5)\) and \(M=(3,4)\).
• Step 2: Solve for \(x\)
• We have \(\frac{2 + x}{2}=3\). Cross - multiply: \(2 + x=6\), then \(x = 4\).
• Step 3: Solve for \(y\)
• We have \(\frac{5 + y}{2}=4\). Cross - multiply: \(5 + y=8\), then \(y = 3\).

1. Find the distance between \(A(3,-1)\) and \(B(7,9)\):

• Answer: \(2\sqrt{29}\)
• Explanation:
• Step 1: Recall the distance formula
• The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \(x_1 = 3,y_1=-1,x_2 = 7,y_2 = 9\).
• Step 2: Calculate \((x_2 - x_1)^2+(y_2 - y_1)^2\)
• \(x_2 - x_1=7 - 3 = 4\), \(y_2 - y_1=9-( - 1)=10\). Then \((x_2 - x_1)^2+(y_2 - y_1)^2=4^2+10^2=16 + 100 = 116\).
• Step 3: Calculate the distance \(d\)
• \(d=\sqrt{116}=\sqrt{4\times29}=2\sqrt{29}\).