Question

practice & problem solving
find the midpoint and length of each segment.

1. $overline{ef}$
2. $overline{fg}$
3. $overline{gh}$
4. $overline{eh}$
5. what are the coordinates of the point that partitions $overline{he}$ in the ratio 2:3?
6. make sense of problems sadie models her neighborhood on a coordinate plane so that her school is at (8, 12) and a store is at (14, 3). what are the coordinates of the point halfway between the school and the store?

Explanation:

Step1: Identify coordinates

Let \(x_1=-2,y_1 = - 3,x_2=-2,y_2 = 1\).
Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(d=\sqrt{(-2+2)^2+(1 + 3)^2}\).
## Step2: Calculate
\(d=\sqrt{0 + 16}=4\).
- **Mid - point**:
## Step1: Use mid - point formula
\(M=(\frac{-2-2}{2},\frac{-3 + 1}{2})\).
## Step2: Simplify
\(M=(-2,-1)\).
- # Answer: Length is 4, mid - point is \((-2,-1)\).
- **Problem 22: For segment \(\overline{FG}\)**
- **Length**:
- # Explanation:
## Step1: Identify coordinates
Let \(x_1=-2,y_1 = 1,x_2=2,y_2 = 4\).
Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(d=\sqrt{(2 + 2)^2+(4 - 1)^2}\).
## Step2: Calculate
\(d=\sqrt{16 + 9}=\sqrt{25}=5\).
- **Mid - point**:
## Step1: Use mid - point formula
\(M=(\frac{-2 + 2}{2},\frac{1+4}{2})\).
## Step2: Simplify
\(M=(0,\frac{5}{2})\).
- # Answer: Length is 5, mid - point is \((0,\frac{5}{2})\).
- **Problem 23: For segment \(\overline{GH}\)**
- **Length**:
- # Explanation:
## Step1: Identify coordinates
Let \(x_1=2,y_1 = 4,x_2=2,y_2 = 1\).
Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(d=\sqrt{(2 - 2)^2+(1 - 4)^2}\).
## Step2: Calculate
\(d=\sqrt{0+9}=3\).
- **Mid - point**:
## Step1: Use mid - point formula
\(M=(\frac{2 + 2}{2},\frac{4+1}{2})\).
## Step2: Simplify
\(M=(2,\frac{5}{2})\).
- # Answer: Length is 3, mid - point is \((2,\frac{5}{2})\).
- **Problem 24: For segment \(\overline{EH}\)**
- **Length**:
- # Explanation:
## Step1: Identify coordinates
Let \(x_1=-2,y_1=-3,x_2=2,y_2 = 1\).
Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(d=\sqrt{(2 + 2)^2+(1 + 3)^2}\).
## Step2: Calculate
\(d=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\).
- **Mid - point**:
## Step1: Use mid - point formula
\(M=(\frac{-2 + 2}{2},\frac{-3+1}{2})\).
## Step2: Simplify
\(M=(0,-1)\).
- # Answer: Length is \(4\sqrt{2}\), mid - point is \((0,-1)\).
2. **For problem 25**:
- Let the coordinates of \(H=(2,1)\) and \(E=(-2,-3)\).
- The section formula for a point \(P(x,y)\) that divides the line - segment joining \((x_1,y_1)\) and \((x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\), where \(m = 2\) and \(n = 3\).
- # Explanation:
## Step1: Calculate \(x\) - coordinate
\(x=\frac{2\times2+3\times(-2)}{2 + 3}=\frac{4-6}{5}=-\frac{2}{5}\).
## Step2: Calculate \(y\) - coordinate
\(y=\frac{2\times1+3\times(-3)}{2 + 3}=\frac{2 - 9}{5}=-\frac{7}{5}\).
- # Answer: The coordinates of the point are \((-\frac{2}{5},-\frac{7}{5})\).
3. **For problem 26**:
- Let the school be at \((8,12)\) and the store be at \((14,3)\).
- The mid - point formula is \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
- # Explanation:
## Step1: Calculate \(x\) - coordinate of mid - point
\(x=\frac{8 + 14}{2}=\frac{22}{2}=11\).
## Step2: Calculate \(y\) - coordinate of mid - point
\(y=\frac{12+3}{2}=\frac{15}{2}=7.5\).
- # Answer: The coordinates of the mid - point are \((11,7.5)\).

Answer:

1. For problems 21 - 24 (finding mid - point and length of line segments):

• First, recall the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) and the mid - point formula \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
• Let's assume the coordinates of the points from the graph:
• Let \(E=( - 2,-3)\), \(F=( - 2,1)\), \(G=(2,4)\), \(H=(2,1)\).
• Problem 21: For segment \(\overline{EF}\)
• Length:
• # Explanation:

Step1: Identify coordinates

Let \(x_1=-2,y_1 = - 3,x_2=-2,y_2 = 1\).
Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(d=\sqrt{(-2+2)^2+(1 + 3)^2}\).

Step2: Calculate

\(d=\sqrt{0 + 16}=4\).

• Mid - point:

Step1: Use mid - point formula

\(M=(\frac{-2-2}{2},\frac{-3 + 1}{2})\).

Step2: Simplify

\(M=(-2,-1)\).

• # Answer: Length is 4, mid - point is \((-2,-1)\).
• Problem 22: For segment \(\overline{FG}\)
• Length:
• # Explanation:

Step1: Identify coordinates

Let \(x_1=-2,y_1 = 1,x_2=2,y_2 = 4\).
Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(d=\sqrt{(2 + 2)^2+(4 - 1)^2}\).

Step2: Calculate

\(d=\sqrt{16 + 9}=\sqrt{25}=5\).

• Mid - point:

Step1: Use mid - point formula

\(M=(\frac{-2 + 2}{2},\frac{1+4}{2})\).

Step2: Simplify

\(M=(0,\frac{5}{2})\).

• # Answer: Length is 5, mid - point is \((0,\frac{5}{2})\).
• Problem 23: For segment \(\overline{GH}\)
• Length:
• # Explanation:

Step1: Identify coordinates

Let \(x_1=2,y_1 = 4,x_2=2,y_2 = 1\).
Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(d=\sqrt{(2 - 2)^2+(1 - 4)^2}\).

Step2: Calculate

\(d=\sqrt{0+9}=3\).

• Mid - point:

Step1: Use mid - point formula

\(M=(\frac{2 + 2}{2},\frac{4+1}{2})\).

Step2: Simplify

\(M=(2,\frac{5}{2})\).

• # Answer: Length is 3, mid - point is \((2,\frac{5}{2})\).
• Problem 24: For segment \(\overline{EH}\)
• Length:
• # Explanation:

Step1: Identify coordinates

Let \(x_1=-2,y_1=-3,x_2=2,y_2 = 1\).
Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(d=\sqrt{(2 + 2)^2+(1 + 3)^2}\).

Step2: Calculate

\(d=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\).

• Mid - point:

Step1: Use mid - point formula

\(M=(\frac{-2 + 2}{2},\frac{-3+1}{2})\).

Step2: Simplify

\(M=(0,-1)\).

• # Answer: Length is \(4\sqrt{2}\), mid - point is \((0,-1)\).

1. For problem 25:

• Let the coordinates of \(H=(2,1)\) and \(E=(-2,-3)\).
• The section formula for a point \(P(x,y)\) that divides the line - segment joining \((x_1,y_1)\) and \((x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\), where \(m = 2\) and \(n = 3\).
• # Explanation:

Step1: Calculate \(x\) - coordinate

\(x=\frac{2\times2+3\times(-2)}{2 + 3}=\frac{4-6}{5}=-\frac{2}{5}\).

Step2: Calculate \(y\) - coordinate

\(y=\frac{2\times1+3\times(-3)}{2 + 3}=\frac{2 - 9}{5}=-\frac{7}{5}\).

• # Answer: The coordinates of the point are \((-\frac{2}{5},-\frac{7}{5})\).

1. For problem 26:

• Let the school be at \((8,12)\) and the store be at \((14,3)\).
• The mid - point formula is \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
• # Explanation:

Step1: Calculate \(x\) - coordinate of mid - point

\(x=\frac{8 + 14}{2}=\frac{22}{2}=11\).

Step2: Calculate \(y\) - coordinate of mid - point

\(y=\frac{12+3}{2}=\frac{15}{2}=7.5\).

• # Answer: The coordinates of the mid - point are \((11,7.5)\).