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Difficulty: MediumSlope of a Line

In the standard (x,y)(x, y) coordinate plane, if a non-horizontal and non-vertical line has a slope of mm, then reflecting this line across the line y=xy = x produces a line with a slope of 1m\frac{1}{m}.

Answer: Answer

Answer

True
Reflecting a point (x,y)(x, y) across the line y=xy = x swaps the coordinates to (y,x)(y, x). For any two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) on the original line, the slope is m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. The corresponding points on the reflected line are (y1,x1)(y_1, x_1) and (y2,x2)(y_2, x_2), yielding a slope of m′=x2−x1y2−y1=1mm' = \frac{x_2 - x_1}{y_2 - y_1} = \frac{1}{m}. Thus, the statement is true.

Step-by-Step Solution

1
Identify the coordinate transformation representing reflection across the line y=xy = x.
Reflecting any point (x,y)(x, y) across the line y=xy = x yields the point (y,x)(y, x).
By definition of reflection across the identity line, the xx-coordinate and yy-coordinate of each point are swapped.
2
Set up the slope formula for the original line using two distinct points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2).
The slope of the original line is m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
The slope of a line is the ratio of vertical change (rise) to horizontal change (run).
3
Determine the slope of the reflected line using the transformed points (y1,x1)(y_1, x_1) and (y2,x2)(y_2, x_2).
The slope of the reflected line is m′=x2−x1y2−y1m' = \frac{x_2 - x_1}{y_2 - y_1}.
Applying the slope formula to the reflected points swaps the numerator and denominator.
4
Relate the slope of the reflected line, m′m', to the slope of the original line, mm.
Since m′=x2−x1y2−y1=1y2−y1x2−x1=1mm' = \frac{x_2 - x_1}{y_2 - y_1} = \frac{1}{\frac{y_2 - y_1}{x_2 - x_1}} = \frac{1}{m}, the slope of the reflected line is 1m\frac{1}{m}.
Taking the reciprocal of the original slope fraction yields the reflected slope fraction, which is valid since the line is non-horizontal (m≠0m \neq 0) and non-vertical (mm is defined).

Key Concept

Reflection of a line across y=xy = x swaps the rise and run of the line, inverting its slope.
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Topics

Slope of a Line

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