Question

Select from the options to correctly fill in the blanks. Two lines that never meet are called lines, and their slopes are . Two lines that meet at right angles are called lines, and their slopes are Options: parallel perpendicular equal negative reciprocals

Short answer

Two lines that never meet are called \(parallel\) lines, and their slopes are \(equal\). Two lines that meet at right angles are called \(perpendicular\) lines, and their slopes are \(negative reciprocals\).

Step-by-step solution

Parallel Lines

Parallel lines are lines in a plane that never intersect or meet each other, no matter how far they are extended. Since they have the same steepness or incline, their slopes are equal.

Perpendicular Lines

Perpendicular lines are lines in a plane that intersect at a right angle (90 degrees). The product of their slopes is -1, which means that their slopes are negative reciprocals of each other. Now we can fill in the blanks with the correct options. Two lines that never meet are called \(parallel\) lines, and their slopes are \(equal\). Two lines that meet at right angles are called \(perpendicular\) lines, and their slopes are \(negative reciprocals\). So, the correct options are: parallel, equal, perpendicular, and negative reciprocals.

Key concepts

Parallel Lines

Imagine standing on a railroad track where the rails run off into the distance; these tracks are like parallel lines—straight paths that never intersect. In mathematics, parallel lines are defined by having the exact same slope, a measure of their steepness. Slope is represented by the letter 'm' in equations, and when two lines are parallel, their 'm' values are equal. This concept is crucial when solving geometric problems involving lines on a graph.
For instance, if we have the line equations y = 2x + 3 and y = 2x - 4, you can observe that the slope of both is 2, confirming they are parallel. When two lines on a graph are parallel, no matter how far you extend them, they will never meet.

Understanding Slope for Parallel Lines

Slope not only determines parallelism but also can be used to verify that lines are indeed parallel. The formula for slope is \( m = \frac{{rise}}{{run}} \), meaning the vertical change over the horizontal change between two points on a line. When working with geometric figures and equations, maintaining an awareness of parallel lines can help avoid mistakes and arrive at the correct solutions.

Perpendicular Lines

While parallel lines are like train tracks, perpendicular lines are more like the letter 'T', where two lines intersect at an exact ninety-degree angle. The slopes of these intersecting lines hold a special relationship: they are negative reciprocals of each other.
Consider two lines that meet at a right angle: if the slope of one line is 'a', the slope of the perpendicular line will be '-1/a'. This relationship ensures the lines intersect just so, creating that perfect right angle, an L-shape, on the graph.

Calculating Slopes of Perpendicular Lines

Let's say we have a line with a slope of 3, and we want to find a line perpendicular to it. The slope of the perpendicular line would be \( -\frac{1}{3} \). Remember, the product of their slopes equals -1, a key point to verify perpendicularity in geometric problems. Recognizing and utilizing this property can lead to successfully solving problems that involve perpendicular lines and their angles.

Slope of a Line

The slope of a line is its angle of incline, which reveals how much a line goes up or down as you move along it. It's essentially the 'steepness' factor of the line on a graph. Calculating slope is simple: for any two points on a line, take the 'rise', or the change in the y-values, and divide it by the 'run', or the change in the x-values.
To use a real-life example, think of a slope as a hill's steepness when you hike. Imagine walking from point A to point B on the hill. The slope tells you how hard the hike is—how much you'll climb (rise) for each step forward (run) you take. In mathematical terms, for two points (x1,y1) and (x2,y2), slope 'm' is calculated by \( m = \frac{{y2 - y1}}{{x2 - x1}} \).

Visualizing Slope

If you're drawing a line on graph paper, start at one point and 'rise' up or down to the level of the next point before 'running' horizontally to it. The direction in which you 'rise' (upwards or downwards) determines if the slope is positive or negative, crucial when analyzing the behavior of lines in Geometry.

Negative Reciprocals

When dealing with slopes of perpendicular lines, we frequently encounter the concept of negative reciprocals. A negative reciprocal is created by taking the opposite (or negative) of the reciprocal of a number. For a slope 'm', its negative reciprocal is '-1/m'. This relationship is the key behind the perpendicularity property.
Why is this important? For instance, in coordinate geometry, if we know a line has a slope of 5, we instantly know that any line perpendicular to it must have a slope of \( -\frac{1}{5} \). Should our line have a negative slope, say -2, a perpendicular line to this would have a positive slope of 1/2. This inverse relationship is a fundamental concept in geometry, linking lines and angles together in a precise and predictable pattern.

Application in Geometric Problems

Recognizing the negative reciprocal allows us to solve for unknowns in geometric problems, especially when confirming the perpendicularity of lines or constructing geometric shapes requiring right angles. Keeping an eye out for these relationships can be a powerful tool in solving complex geometry problems.