Unit 5, Lesson 6
Perpendicular Lines in the Plane
Integrated Math 1
6.1
Warm-up
The image shows quadrilateral .
Apply the transformation rule to quadrilateral . What is the effect of the transformation rule?
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The image shows quadrilateral .
Apply the transformation rule to quadrilateral . What is the effect of the transformation rule?
The diagram shows triangle and its image, triangle , under a 90-degree rotation counterclockwise using the origin as the center.
Since the rotation was through 90 degrees, all line segments in the image are perpendicular to the corresponding segments in the original triangle. For example, segment is horizontal, while segment is vertical.
Look at segments and , which, like the other pairs of segments, are perpendicular. The slope of segment is , while the slope of segment is . Notice the relationship between the slopes: They are reciprocals of one another, and have opposite signs. The product of the slopes, , is -1. As long as perpendicular lines are not horizontal or vertical, their slopes will be opposite reciprocals and have a product of -1.
We can use this fact to help write equations of lines. For example, try writing the equation of a line that passes through the point and is perpendicular to a line, , represented by . The slope of line is 3. The slope of any line perpendicular to line is the opposite reciprocal of 3, or . Substitute the point and the slope into the point-slope form to get the equation .
Two numbers that multiply to equal 1 are reciprocals.
If is a rational number that is not 0, then the reciprocal of is the number .