Question

vertical segments

1. practice: making mental images and summarizing

the endpoints of a segment are (0, 10) and (0, 20).
how do you know it is a vertical segment?

1. practice: organizing information and summarizing

fill in the blanks to complete the list.
midpoint of vertical segment

• __ -coordinate = __ -coordinate of endpoints
• __ -coordinate = average of __ -coordinates of endpoints

1. practice: making mental images

find the midpoint of each segment. show your work.

1. segment endpoints: (0, 0) and (0, 20)

midpoint = ____

1. segment endpoints: (0, -10) and (0, 20)

midpoint = ____
diagonal segments

1. practice: making mental images and summarizing

the endpoints of a segment are (10, 3) and (20, 17).
how do you know it is a diagonal segment?

Explanation:

Step1: Determine vertical - segment property

In a 2 - D coordinate system, for a segment with endpoints \((x_1,y_1)\) and \((x_2,y_2)\), if \(x_1 = x_2\), the segment is vertical. For endpoints \((0,10)\) and \((0,20)\), \(x_1=0\) and \(x_2 = 0\), so it is a vertical segment.

Step2: Mid - point formula for vertical segment

The mid - point of a segment with endpoints \((x_1,y_1)\) and \((x_2,y_2)\) has coordinates \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). For a vertical segment (\(x_1=x_2\)), the \(x\) - coordinate of the mid - point is the same as the \(x\) - coordinate of the endpoints. The \(y\) - coordinate of the mid - point is the average of the \(y\) - coordinates of the endpoints. So, \(x\) - coordinate=\(x\) - coordinate of endpoints; \(y\) - coordinate = average of \(y\) - coordinates of endpoints.

Step3: Find mid - point of segment \((0,0)\) and \((0,20)\)

Using the mid - point formula \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\), with \(x_1 = 0,x_2 = 0,y_1 = 0,y_2 = 20\). The \(x\) - coordinate of the mid - point is \(\frac{0 + 0}{2}=0\), and the \(y\) - coordinate is \(\frac{0+20}{2}=10\). So the mid - point is \((0,10)\).

Step4: Find mid - point of segment \((0, - 10)\) and \((0,20)\)

Using the mid - point formula \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\), with \(x_1 = 0,x_2 = 0,y_1=-10,y_2 = 20\). The \(x\) - coordinate of the mid - point is \(\frac{0 + 0}{2}=0\), and the \(y\) - coordinate is \(\frac{-10 + 20}{2}=5\). So the mid - point is \((0,5)\).

Step5: Determine diagonal - segment property

In a 2 - D coordinate system, for a segment with endpoints \((x_1,y_1)\) and \((x_2,y_2)\), if \(x_1
eq x_2\) and \(y_1
eq y_2\), the segment is diagonal. For endpoints \((10,3)\) and \((20,17)\), \(x_1 = 10,x_2 = 20,y_1 = 3,y_2 = 17\), since \(x_1
eq x_2\) and \(y_1
eq y_2\), it is a diagonal segment.

Answer:

1. The \(x\) - coordinates of the endpoints \((0,10)\) and \((0,20)\) are the same.
2. \(x\), \(x\); \(y\), \(y\)
3. 1. \((0,10)\); 2. \((0,5)\)
4. The \(x\) - coordinates and \(y\) - coordinates of the endpoints \((10,3)\) and \((20,17)\) are different.