Question

use the questions below to keep track of key concepts from this lessons study activity. 1) practice: summarizing fill in the blanks in the definition. the mid - point of a segment is the point ______ between its two ____. horizontal segments 2) practice: making mental images and summarizing the endpoints of a segment are (10, 0) and (20, 0). how do you know it is a horizontal segment? ____ 3) practice: organizing information and summarizing fill in the blanks to complete the list. midpoint of horizontal segment - ____ - coordinate = average of ____ - coordinates of endpoints - ____ - coordinate = ____ - coordinates of endpoints 4) practice: making mental images find the mid - point of each segment. show your work. 1. segment endpoints: (0, 0) and (20, 0) midpoint = ____ 2. segment endpoints: (-10, 0) and (20, 0) midpoint = ______

Explanation:

Step1: Recall mid - point definition

The midpoint of a segment is the point equidistant between its two endpoints.

Step2: Identify horizontal segment property

For a segment with endpoints \((x_1,y_1)\) and \((x_2,y_2)\), if \(y_1 = y_2\), it is a horizontal segment. Given endpoints \((10,0)\) and \((20,0)\), since the \(y\) - coordinates are the same (\(y = 0\) for both), it is a horizontal segment.

Step3: Recall mid - point formula for horizontal segment

For a horizontal segment, the \(x\) - coordinate of the mid - point is the average of the \(x\) - coordinates of the endpoints, and the \(y\) - coordinate of the mid - point is the same as the \(y\) - coordinates of the endpoints. So, \(x\) - coordinate = average of \(x\) - coordinates of endpoints, \(y\) - coordinate = \(y\) - coordinates of endpoints.

Step4: Calculate mid - point for \((0,0)\) and \((20,0)\)

The formula for the mid - point \((x_m,y_m)\) of a segment with endpoints \((x_1,y_1)\) and \((x_2,y_2)\) is \(x_m=\frac{x_1 + x_2}{2}\) and \(y_m=\frac{y_1 + y_2}{2}\). For \((x_1 = 0,y_1 = 0)\) and \((x_2 = 20,y_2 = 0)\), \(x_m=\frac{0 + 20}{2}=10\) and \(y_m=\frac{0+0}{2}=0\). So the mid - point is \((10,0)\).

Step5: Calculate mid - point for \((- 10,0)\) and \((20,0)\)

Using the mid - point formula \(x_m=\frac{-10 + 20}{2}=\frac{10}{2}=5\) and \(y_m=\frac{0 + 0}{2}=0\). So the mid - point is \((5,0)\).

Answer:

1. The midpoint of a segment is the point equidistant between its two endpoints.
2. Since the \(y\) - coordinates of the endpoints \((10,0)\) and \((20,0)\) are the same (\(y = 0\)), it is a horizontal segment.
3. \(x\) - coordinate = average of \(x\) - coordinates of endpoints, \(y\) - coordinate = \(y\) - coordinates of endpoints

4.

• For segment endpoints \((0,0)\) and \((20,0)\), midpoint = \((10,0)\)
• For segment endpoints \((-10,0)\) and \((20,0)\), midpoint = \((5,0)\)