Distance And Midpoint Worksheet Answers Explained Simply
Table of Contents :
Understanding distance and midpoint calculations is essential for solving various mathematical problems, especially in geometry. The concepts help us find the separation between two points and the exact middle point of a line segment. In this article, we will break down how to find distance and midpoint with explanations and examples that are easy to understand.
What is Distance?
The distance between two points in a Cartesian plane can be calculated using the distance formula. If you have two points, ( A(x_1, y_1) ) and ( B(x_2, y_2) ), the distance ( d ) is given by the formula:
[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]
Example of Distance Calculation
Let’s say we have two points, ( A(1, 2) ) and ( B(4, 6) ).
Using the distance formula:
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Step 1: Substitute the coordinates into the formula:
[ d = \sqrt{(4 - 1)^2 + (6 - 2)^2} ]
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Step 2: Calculate inside the parentheses:
[ d = \sqrt{(3)^2 + (4)^2} ]
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Step 3: Square the values:
[ d = \sqrt{9 + 16} ]
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Step 4: Add the results:
[ d = \sqrt{25} ]
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Step 5: Take the square root:
[ d = 5 ]
So, the distance between points ( A(1, 2) ) and ( B(4, 6) ) is 5 units. 📏
What is Midpoint?
The midpoint of a line segment is the point that divides the segment into two equal parts. It can be calculated using the midpoint formula. For points ( A(x_1, y_1) ) and ( B(x_2, y_2) ), the midpoint ( M ) is given by:
[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ]
Example of Midpoint Calculation
Let’s find the midpoint of the same points ( A(1, 2) ) and ( B(4, 6) ).
Using the midpoint formula:
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Step 1: Substitute the coordinates into the formula:
[ M = \left( \frac{1 + 4}{2}, \frac{2 + 6}{2} \right) ]
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Step 2: Calculate each coordinate:
[ M = \left( \frac{5}{2}, \frac{8}{2} \right) ]
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Step 3: Simplify the coordinates:
[ M = \left( 2.5, 4 \right) ]
Thus, the midpoint between points ( A(1, 2) ) and ( B(4, 6) ) is ( M(2.5, 4) ). 📍
Summary Table of Distance and Midpoint
To summarize the formulas and methods we used, let’s take a look at this table:
<table> <tr> <th>Concept</th> <th>Formula</th> <th>Example with Points (1, 2) & (4, 6)</th> </tr> <tr> <td>Distance</td> <td>d = √((x2 - x1)<sup>2</sup> + (y2 - y1)<sup>2</sup>)</td> <td>d = 5 units</td> </tr> <tr> <td>Midpoint</td> <td>M = ((x1 + x2)/2, (y1 + y2)/2)</td> <td>M = (2.5, 4)</td> </tr> </table>
Important Notes
- Always ensure that the coordinates are in the correct order when using the formulas.
- Practice with different points to master these concepts.
Practice Problems
- Find the distance between points ( C(2, 3) ) and ( D(6, 7) ).
- Find the midpoint between points ( E(0, 0) ) and ( F(8, 6) ).
Feel free to use the distance and midpoint formulas to check your answers!
Conclusion
Distance and midpoint calculations are straightforward once you understand the formulas and the steps involved. With a bit of practice, you will become proficient in finding distances and midpoints of various points in a Cartesian plane. 🧮 Keep practicing, and these concepts will become second nature to you!