Question

If triangle XYZ is reflected across the line y = 1 to create triangle X'Y'Z', what is the ordered pair of X'? The original X coordinate is (4,5)

Answer 1

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Answer 2

By Definition, the Reflection of a Point A in a line l is the point A' that has the following Property (P) :
(P) : The Line l the Perpendicular Bisector of the Line Segment AA' .

Let X'(a,b) be the reflection of the point X(4,5) in the line l : y=1.
So, XX' must be perpendicular to l.
Note that the line l is Horizontal, i.e., parallel to the X-Axis.
Hence, XX' (being perpendicular to a Horizontal line l) must be Vertical, or, parallel to the Y-Axis.
This means that the equation (eqn.) of XX' must be of the form x=k (k, constant).
Knowing that XX' passes through X(4,5), the eqn. of XX' must be XX' : x=4. :. a=4.

Next, l bisects XX', the mid-point of Segment XX' must lie on l.
But, X(4,5), X'(a,b)=(4,b); so, the mid-point of XX' is ((4+4)/2,(5+b)/2).
This mid-point lies on l : y=1.
:. (5+b)/2=1, or, b=-3.
So the requred reflection of X(4,5) is X'(a,b)=X'(4,-3).

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