Distance And Midpoint Worksheet Answers Explained
Table of Contents :
Understanding distance and midpoint calculations is crucial for mastering geometric concepts. In this article, we will explore distance and midpoint formulas, provide solutions to common problems, and present a worksheet with answers explained. By the end, you’ll have a thorough grasp of these concepts, which will enhance your problem-solving skills in geometry.
What are Distance and Midpoint?
Distance
The distance between two points in a coordinate plane can be determined using the distance formula derived from the Pythagorean theorem. The formula is:
Distance Formula:
[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
]
Where:
- (d) = distance between two points
- ((x_1, y_1)) and ((x_2, y_2)) are the coordinates of the points.
Midpoint
The midpoint of a line segment is the point that is exactly halfway between two endpoints. The midpoint formula is as follows:
Midpoint Formula:
[
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
]
Where:
- (M) = midpoint
- ((x_1, y_1)) and ((x_2, y_2)) are the coordinates of the endpoints.
Common Problems in Worksheets
To better understand these concepts, let's look at an example worksheet with specific problems, followed by solutions that explain the answers in detail.
Example Problems
- Find the distance between the points A(2, 3) and B(5, 7).
- Determine the midpoint of the line segment connecting points C(-1, -4) and D(3, 2).
- Calculate the distance between the points E(-3, 1) and F(4, -5).
- Find the midpoint of the line segment connecting points G(0, 0) and H(8, 6).
Answers Explained
Now, let's tackle these problems step by step to understand the solutions clearly.
Problem 1: Distance between A(2, 3) and B(5, 7)
Using the distance formula:
-
Identify the coordinates:
- ( (x_1, y_1) = (2, 3) )
- ( (x_2, y_2) = (5, 7) )
-
Apply the distance formula: [ d = \sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 ]
Answer: The distance is 5 units. 🚶♂️
Problem 2: Midpoint of C(-1, -4) and D(3, 2)
Using the midpoint formula:
-
Identify the coordinates:
- ( (x_1, y_1) = (-1, -4) )
- ( (x_2, y_2) = (3, 2) )
-
Apply the midpoint formula: [ M = \left(\frac{-1 + 3}{2}, \frac{-4 + 2}{2}\right) = \left(\frac{2}{2}, \frac{-2}{2}\right) = (1, -1) ]
Answer: The midpoint is (1, -1). 🔗
Problem 3: Distance between E(-3, 1) and F(4, -5)
Using the distance formula:
-
Identify the coordinates:
- ( (x_1, y_1) = (-3, 1) )
- ( (x_2, y_2) = (4, -5) )
-
Apply the distance formula: [ d = \sqrt{(4 - (-3))^2 + (-5 - 1)^2} = \sqrt{(4 + 3)^2 + (-6)^2} = \sqrt{7^2 + 6^2} = \sqrt{49 + 36} = \sqrt{85} \approx 9.22 ]
Answer: The distance is approximately 9.22 units. 📏
Problem 4: Midpoint of G(0, 0) and H(8, 6)
Using the midpoint formula:
-
Identify the coordinates:
- ( (x_1, y_1) = (0, 0) )
- ( (x_2, y_2) = (8, 6) )
-
Apply the midpoint formula: [ M = \left(\frac{0 + 8}{2}, \frac{0 + 6}{2}\right) = \left(4, 3\right) ]
Answer: The midpoint is (4, 3). ⚖️
Summary Table of Problems and Solutions
<table> <tr> <th>Problem</th> <th>Solution</th> </tr> <tr> <td>Distance between A(2, 3) and B(5, 7)</td> <td>5 units</td> </tr> <tr> <td>Midpoint of C(-1, -4) and D(3, 2)</td> <td>(1, -1)</td> </tr> <tr> <td>Distance between E(-3, 1) and F(4, -5)</td> <td>≈ 9.22 units</td> </tr> <tr> <td>Midpoint of G(0, 0) and H(8, 6)</td> <td>(4, 3)</td> </tr> </table>
Final Thoughts
Understanding how to calculate distance and midpoint in geometry is an essential skill. By practicing problems like those mentioned above, you will improve your ability to apply these formulas in various contexts. Whether you are preparing for an exam or simply brushing up on your math skills, mastering distance and midpoint will serve you well in your academic journey. Happy learning! 📚