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2. XY has an endpoint X at (2,– 4). If the midpoint of XY is (– 3,– 1), what are the coordinates of Y? Y:

2. XY has an endpoint X at (2,– 4). If the midpoint of XY is (– 3,– 1), what are the coordinates of Y? Y:

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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### Midpoint Calculation Problem #### Problem Statement: Line segment \( \overline{XY} \) has an endpoint \( X \) at \((2, -4)\). If the midpoint of \( \overline{XY} \) is \((-3, -1)\), what are the coordinates of \( Y \)? **Solution Box:** \( Y: \_\_\_\_\_\_\_\_\_\_\_\_ \) #### Explanation of the Problem: This problem involves finding the coordinates of the other endpoint \( Y \) of a line segment when one endpoint \( X \) and the midpoint are given. The coordinates of \( X \) are \((2, -4)\), and the coordinates of the midpoint are \((-3, -1)\). #### Steps to Solve: 1. **Understand the Midpoint Formula:** The midpoint \((M)\) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] 2. **Set Up the Equation:** Given the midpoint \((-3, -1)\) and one endpoint \( X(2, -4) \), set up the equations for the x and y coordinates of the midpoint. 3. **Solve for \( x_2 \) and \( y_2 \) (coordinates of \( Y \)):** \[ \left( \frac{2 + x_2}{2}, \frac{-4 + y_2}{2} \right) = (-3, -1) \] Equate the x-coordinates: \[ \frac{2 + x_2}{2} = -3 \\ 2 + x_2 = -6 \\ x_2 = -6 - 2 \\ x_2 = -8 \] Equate the y-coordinates: \[ \frac{-4 + y_2}{2} = -1 \\ -4 + y_2 = -2 \\ y_2 = -2 + 4 \\ y_2 = 2 \] 4. **
Transcribed Image Text:### Midpoint Calculation Problem #### Problem Statement: Line segment \( \overline{XY} \) has an endpoint \( X \) at \((2, -4)\). If the midpoint of \( \overline{XY} \) is \((-3, -1)\), what are the coordinates of \( Y \)? **Solution Box:** \( Y: \_\_\_\_\_\_\_\_\_\_\_\_ \) #### Explanation of the Problem: This problem involves finding the coordinates of the other endpoint \( Y \) of a line segment when one endpoint \( X \) and the midpoint are given. The coordinates of \( X \) are \((2, -4)\), and the coordinates of the midpoint are \((-3, -1)\). #### Steps to Solve: 1. **Understand the Midpoint Formula:** The midpoint \((M)\) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] 2. **Set Up the Equation:** Given the midpoint \((-3, -1)\) and one endpoint \( X(2, -4) \), set up the equations for the x and y coordinates of the midpoint. 3. **Solve for \( x_2 \) and \( y_2 \) (coordinates of \( Y \)):** \[ \left( \frac{2 + x_2}{2}, \frac{-4 + y_2}{2} \right) = (-3, -1) \] Equate the x-coordinates: \[ \frac{2 + x_2}{2} = -3 \\ 2 + x_2 = -6 \\ x_2 = -6 - 2 \\ x_2 = -8 \] Equate the y-coordinates: \[ \frac{-4 + y_2}{2} = -1 \\ -4 + y_2 = -2 \\ y_2 = -2 + 4 \\ y_2 = 2 \] 4. **
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