In coordinate geometry, we often need to find the reflection of a point across a line. Given point P at coordinates 1, 3 and line L with equation x plus y minus 2 equals 0, we want to find the symmetric point P prime. The reflection creates a perpendicular relationship where the original line bisects the segment connecting P and P prime.
The reflection formula provides a direct method to calculate the symmetric point. For any point P with coordinates x naught, y naught, and a line with equation A x plus B y plus C equals zero, the reflected point P prime has coordinates given by these formulas. The x coordinate of the reflection is x naught minus two A times the quantity A x naught plus B y naught plus C, all divided by A squared plus B squared. Similarly, the y coordinate uses the same pattern with coefficient B.
Let's work through the calculation step by step. First, we identify our given information: point P at coordinates 1, 3, and line L with equation x plus y minus 2 equals zero. From the line equation, we have A equals 1, B equals 1, and C equals negative 2. Next, we calculate D by substituting the point coordinates into the line equation: D equals 1 times 1 plus 1 times 3 plus negative 2, which equals 2. Then we calculate E as A squared plus B squared: E equals 1 squared plus 1 squared, which equals 2.
Now we complete the calculation. For the x coordinate of the reflection, we use the formula x prime equals x naught minus two A D over E. Substituting our values: x prime equals 1 minus 2 times 1 times 2 over 2, which equals 1 minus 2, giving us negative 1. For the y coordinate, y prime equals y naught minus two B D over E. This gives us y prime equals 3 minus 2 times 1 times 2 over 2, which equals 3 minus 2, giving us 1. Therefore, the reflection of point P at coordinates 1, 3 across the line x plus y minus 2 equals zero is P prime at coordinates negative 1, 1.
Let's verify our solution and summarize the method. First, we check that the midpoint M at coordinates 0, 2 lies on the line: substituting into x plus y minus 2 equals zero gives us 0 plus 2 minus 2 equals 0, which is correct. Second, we verify that line PP prime is perpendicular to line L: the direction vector of PP prime is negative 2, negative 2, which is parallel to the normal vector 1, 1 of line L, confirming perpendicularity. In summary, the reflection formula provides a systematic way to find the symmetric point of any given point across any line in the coordinate plane.