Reflection of a Point Under the Line y -2
Reflection of a Point Under the Line y -2
Understanding geometric transformations is crucial in many fields, including computer graphics and physics. One of the fundamental transformations is the reflection of a point over a line. In this article, we will explore the reflection of a point (5, 7) over the line y -2. We will use coordinate geometry and the reflection formula to find the coordinates of the reflected point.
Step-by-Step Solution
To find the reflection of a point (x, y) over the line y -2, we can use the following steps:
Identify the line of reflection: y -2. Use the reflection formula: x - X/a (y - Y)/b -2aX/bY/c/a^2b^2. Plug in the values: a 0, b 1, c -2, X 5, Y 7.Calculations and Explanation
Let's start with the given point (5, 7) and the line y -2. We need to determine the coordinates of the reflected point (X, Y).
The x-coordinate of the reflected point is constant since the line of reflection is parallel to the x-axis. Therefore, x X 5.
Now, let's calculate the y-coordinate using the reflection formula:
Given the line y -2, the formula for the reflection becomes:
x - 5 0
X 5
y - 7 -2(1)(7)/1^2(1^2)
Since b 1 and a 0, we can simplify this to:
y - 7 -14
Therefore, y -7 14 11. The y-coordinate of the reflected point is 11.
Thus, the coordinates of the reflected point are (5, -11).
Verification and Graphical Representation
To verify our calculations, we can draw a graph and check the reflection.
Graph paper representation:
Plot the original point (5, 7). Draw the line y -2. Reflect the point (5, 7) over the line y -2. Check if the reflected point is (5, -11).A quick sketch of the graph will show that the reflection is correct.
Conclusion
The reflection of the point (5, 7) over the line y -2 is the point (5, -11). This is a classic example of how coordinate geometry can be applied to solve geometric problems.