Geometry Worksheet 1.3: Distance & Midpoints Answer Key
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Geometry is a fascinating branch of mathematics that focuses on shapes, sizes, relative positions of figures, and properties of space. One of the essential aspects of geometry involves understanding distance and midpoints, concepts that lay the groundwork for more advanced topics in the field. In this article, we will dive deep into the understanding of distance and midpoints, with a particular focus on the content typically covered in a Geometry Worksheet 1.3. We'll also provide the answer key to enhance your learning experience. Let's get started! 📐✨
Understanding Distance and Midpoints
What is Distance?
Distance refers to the length between two points in a geometric space. In a two-dimensional plane, the distance between two points ((x_1, y_1)) and ((x_2, y_2)) can be calculated using the Distance Formula:
[ d = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2} ]
Where:
- (d) = distance
- ((x_1, y_1)) and ((x_2, y_2)) are the coordinates of the two points.
What are Midpoints?
A midpoint is a point that divides a line segment into two equal parts. If you have two points ((x_1, y_1)) and ((x_2, y_2)), the midpoint (M) can be calculated with the 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.
Real-life Applications
Understanding distance and midpoints has various applications in real life, from navigation systems that compute the shortest distance between locations to architecture that relies on accurate measurements for design purposes. 🚀
Geometry Worksheet 1.3: Distance & Midpoints
In a typical Geometry Worksheet 1.3, you would encounter problems that require you to apply the distance and midpoint formulas. Let’s look at some examples to illustrate the concepts:
Example Problems
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Find the Distance: Calculate the distance between the points (A(1, 2)) and (B(4, 6)).
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Find the Midpoint: Determine the midpoint of the segment connecting points (C(2, 3)) and (D(4, 7)).
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Mixed Problem: Given points (E(-1, -1)) and (F(3, 3)), find both the distance and midpoint.
Solution Steps
Here’s how we solve the above examples:
Problem 1: Finding the Distance
Using the Distance Formula:
[ d = \sqrt{{(4 - 1)}^2 + {(6 - 2)}^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 ]
Problem 2: Finding the Midpoint
Using the Midpoint Formula:
[ M = \left( \frac{{2 + 4}}{2}, \frac{{3 + 7}}{2} \right) = \left( 3, 5 \right) ]
Problem 3: Mixed Problem
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Finding Distance: [ d = \sqrt{{(3 - (-1))}^2 + {(3 - (-1))}^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.66 ]
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Finding Midpoint: [ M = \left( \frac{{-1 + 3}}{2}, \frac{{-1 + 3}}{2} \right) = \left( 1, 1 \right) ]
Summary of Solutions
To make it easier for reference, here’s a summary in table format:
<table> <tr> <th>Problem</th> <th>Distance (d)</th> <th>Midpoint (M)</th> </tr> <tr> <td>1: A(1, 2) and B(4, 6)</td> <td>5</td> <td>N/A</td> </tr> <tr> <td>2: C(2, 3) and D(4, 7)</td> <td>N/A</td> <td>(3, 5)</td> </tr> <tr> <td>3: E(-1, -1) and F(3, 3)</td> <td>5.66</td> <td>(1, 1)</td> </tr> </table>
Important Notes
"Mastering distance and midpoint calculations is vital for solving many geometry problems and is a foundational skill for more advanced mathematics." 📏
Conclusion
As you work through Geometry Worksheet 1.3, remember the significance of the distance and midpoint formulas. They not only provide solutions to specific problems but also enhance your understanding of geometric principles and their applications. Practice makes perfect, so keep honing these skills, and soon you will find yourself confidently solving a wide range of geometry problems! Happy studying! 📚