Find AB Given A (5, -3) And B (-11, -5).
When working with points in a coordinate plane, one of the fundamental tasks is to determine the distance between two points. This process involves understanding the coordinates of the points and applying the distance formula derived from the Pythagorean theorem. In this article, we will explore in detail how to find the length of the segment AB, given the coordinates of points A and B, specifically A(5, -3) and B(-11, -5). We will break down the process step by step, discuss the relevant formulas, and provide examples to clarify the method.
Understanding the Coordinates of Points A and B
Coordinates of Point A
- Point A is located at (5, -3)
- The x-coordinate of A is 5
- The y-coordinate of A is -3
Coordinates of Point B
- Point B is located at (-11, -5)
- The x-coordinate of B is -11
- The y-coordinate of B is -5
Understanding these coordinates is essential because the distance formula relies on the differences in the x-coordinates and y-coordinates of the two points.
The Distance Formula
Derivation of the Distance Formula
The distance between two points A(x₁, y₁) and B(x₂, y₂) can be thought of as the hypotenuse of a right-angled triangle formed by the differences in their x and y coordinates. According to the Pythagorean theorem:\[
AB = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}
\]
This formula calculates the straight-line distance between the points in the coordinate plane.
Applying the Formula
To find AB, substitute the coordinates of points A and B into the formula:\[
AB = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}
\]
Where:
- \( x1 = 5 \), \( y1 = -3 \)
- \( x2 = -11 \), \( y2 = -5 \)
So, the specific calculation becomes:
\[
AB = \sqrt{(-11 - 5)^2 + (-5 - (-3))^2}
\]
Now, let's proceed step by step.
Calculating the Distance Step by Step
Step 1: Find the difference in x-coordinates
\[ x2 - x1 = -11 - 5 = -16 \]Step 2: Find the difference in y-coordinates
\[ y2 - y1 = -5 - (-3) = -5 + 3 = -2 \]Step 3: Square the differences
\[ (-16)^2 = 256 \] \[ (-2)^2 = 4 \]Step 4: Sum the squared differences
\[ 256 + 4 = 260 \]Step 5: Take the square root to find the distance
\[ AB = \sqrt{260} \]Final Calculation and Result
Simplifying the Square Root
The square root of 260 can be simplified by factoring out perfect squares:\[
\sqrt{260} = \sqrt{4 \times 65} = \sqrt{4} \times \sqrt{65} = 2 \sqrt{65}
\]
Therefore, the exact length of AB is:
\[
AB = 2 \sqrt{65}
\]
If a decimal approximation is desired, we evaluate \(\sqrt{65}\):
\[
\sqrt{65} \approx 8.062
\]
So,
\[
AB \approx 2 \times 8.062 = 16.124
\]
Result: The length of segment AB is approximately 16.124 units.
Additional Considerations
Using a Calculator
For quick approximate results, a calculator can be used to compute \(\sqrt{260}\) directly, which yields approximately 16.124.Understanding the Significance
Knowing how to find the distance between two points is fundamental in geometry, navigation, and many applied fields such as physics, engineering, and computer graphics.Summary of the Process
To summarize, the key steps to find the length AB given points A and B are:
- Identify the coordinates of A and B.
- Calculate the differences in the x-coordinates and y-coordinates.
- Square these differences.
- Sum the squared differences.
- Take the square root of the sum to get the distance.
This systematic approach ensures accuracy and clarity in solving problems involving distances in the coordinate plane.
Practice Problems
Try These on Your Own
- Find the distance between C(2, 4) and D(10, -2).
- Determine the length of segment EF, given E(-3, 7) and F(4, 1).
- Calculate the distance between points G(0, 0) and H(5, 12).
- \( \sqrt{(10 - 2)^2 + (-2 - 4)^2} = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \)
- \( \sqrt{(4 - (-3))^2 + (1 - 7)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85} \approx 9.22 \)
- \( \sqrt{(5 - 0)^2 + (12 - 0)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \)
Conclusion
Finding the distance between two points in a coordinate plane is a fundamental skill in geometry that combines understanding of coordinate systems with the application of the Pythagorean theorem. By carefully substituting the points' coordinates into the distance formula, performing the necessary calculations step by step, and simplifying the result, you can accurately determine the length of the segment connecting any two points. Whether working with the exact radical form or a decimal approximation, this method provides a clear and reliable way to approach such problems.
