In the rectangular coordinate system, the line y = -x is the
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05 Oct 2020, 08:49
Concept: when an Original Point is Reflected over a LINE:
(1) the Original Point and the new Reflected “Mirror Image” Point are EQUI-DISTANT from the Line
AND
(2) this “Mirror Line” over which the Original Point was Reflected is the Perpendicular Bisector of the Line Segment connecting the 2 Points
Concept 2: when a Point is Reflected over the Line given by the Equation: y = -(x)
Original Point (X , Y)
——> become Reflected “Mirror” Point (-Y , -X)
In other words, when you Reflect a Point over the Line: y = -(x)
The Coordinates get SWITCHED and NEGATED
Thus, Point R (-3 , 1) Reflected over the Line: y = -(x) ———> results in the Mirror Point S with Coordinates of (-1 , + 3)
If you join a Line connecting Point R and its Reflected Point S, you can see that Line: y = -(x) is the PERPENDICULAR Bisector of Line Segment RS
Proof:
(1st) Point R (-3 , 1)
(a) Plot the Point on Line: y = -(x) that has the SAME X Coordinate of -3
(-3 , 3) will be on Line y = -(x). Call this Point A.
The Vertical Distance from Point R to this Point A on Line y = -(x) is exactly 2 Units
(b)Plot the Point on Line: y = -(x) that has the SAME Y Coordinate of +1
(-1 , 1) will be the Point on Line y = -(x). Call this Point B.
The Horizontal Distance from Point R to this Point B on Line y = -(x) is exactly 2 Units
(2nd) Reflected Image Point S (-1 , 3)
Perform the same logical exercise and connect Point S to Points A and B on the Mirror Line of : y = -(x)
You will see that this is a 2 by 2 Square. And in Squares, the 2 Diagonals from Vertex R to Vertex S and From Vertex A to Vertex B are Perpendicular Bisectors
Summary: this means Line y = -(x) will be the Perpendicular Bisector of the Line Segment dawn from Point R (-3 , 1) to Point S (-1 , 3)
Thus, Point S has the coordinates of (-1 , 3)
Lastly, we need the Y-Axis (given by Line Equation: x = 0) to be the Perpendicular Bisector of the Line Segment ST, with unknown Coordinates for Point T.
Using similar logic, if we Reflect Point R over the Y-Axis (x = 0) we will get the Reflected Image Point T. Just like above, the Y- Axis will be the Perpendicular Bisector of Line Segment ST
concept 3: when we Reflect an Original Point (X , Y) over the Y-Axis:
Original Point (X , Y) becomes ——> Reflected Image Point (-X , Y)
In other words, we keep the SAME Y Coordinate and NEGATE the X Coordinate.
Thus, Point T will be at (+1 , 3)
If you plot both these Points, it is easy to see that the Y Axis is the Perpendicular Bisector of Line Segment ST.
Answer -D-
(1 , 3) are Coordinates of Point T
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