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Perpendicular Lines and Slopes

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Perpendicular lines intersect at a right angle, forming four 90-degree angles. This text explores their slopes, which are negative reciprocals, and how to calculate and formulate equations for lines perpendicular to a given line. Practical applications include building design and geometric problem-solving, demonstrating the importance of understanding these fundamental concepts in geometry.

Defining Perpendicular Lines

Perpendicular lines are an essential concept in geometry, defined as two lines that intersect at a right angle, which is an angle of exactly 90 degrees. This perpendicularity is denoted by the symbol \( AB \perp CD \), signifying that line segment AB is perpendicular to line segment CD. At the point of intersection, four right angles are formed, each measuring 90 degrees. Perpendicular lines are not only a theoretical construct but also have practical applications in everyday life, such as in the design of buildings, where walls intersect floors at right angles, and in various symbols, including the cross on a first aid kit.
Carpenter's square on a wooden workbench, showcasing perpendicular arms with a soft shadow, amidst sawdust, highlighting precision in woodworking.

Slope of Perpendicular Lines

The slope, or gradient, of a line quantifies its steepness and direction. It is a key concept when examining perpendicular lines, as the slope determines the angle a line makes with the horizontal. Mathematically, the slope is represented by 'm' in the linear equation \( y = mx + b \), where 'b' is the y-intercept. A unique characteristic of perpendicular lines is that their slopes are negative reciprocals of one another. If one line has a slope of \( m_1 \), then a line perpendicular to it will have a slope of \( m_2 \), such that \( m_1 \cdot m_2 = -1 \). This inverse relationship is essential for establishing the perpendicularity between two lines.

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00

Definition of slope in a line

Slope quantifies line steepness/direction, represented by 'm' in y=mx+b.

01

Meaning of 'b' in linear equation

'b' is the y-intercept, where line crosses y-axis.

02

Slope of perpendicular lines relationship

Perpendicular lines have slopes that are negative reciprocals (m1 * m2 = -1).

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