Understanding the Concept of Parallel Lines in Geometry

Geometry is a branch of mathematics that deals with the properties and relationships of points, lines, surfaces, and solids. One fundamental concept in this field is that of parallel lines. Parallel lines are lines in a plane that never intersect, no matter how far they are extended. To find the equation of a line that is parallel to a given line and passes through a specific point, we need to understand the principles of slope and the slope-intercept form of a line.

Step-by-Step Guide

Let's explore the detailed steps in finding the equation of a line parallel to a given line and passing through a specific point. We will start with the given line and point and proceed through the necessary calculations.

Step 1: Determine the Slope of the Original Line

The first step is to rewrite the given equation in slope-intercept form, which is y mx b. This form allows us to easily identify the slope m. For the given line 3x - 4y 6 0, we can rearrange it as follows:

Starting with the original equation: 3x - 4y 6 0

Rearrange to isolate y: -4y -3x - 6

Divide by -4: y frac{3}{4}x frac{3}{2}

From this, we see that the slope m of the line is frac{3}{4}.

Step 2: Use the Slope for the Parallel Line

Parallel lines have the same slope. Therefore, the slope of the line we want to find will also be frac{3}{4}.

Step 3: Use the Point-Slope Form of the Line

We can use the point-slope form of a line equation, which is y - y? m(x - x?). Here, (x?, y?) is the point (2, -3), and m is the slope frac{3}{4}.

Using the point-slope form: y - (-3) frac{3}{4}(x - 2)

Adding the -3: y 3 frac{3}{4}x - frac{3}{2}

Subtracting 3 from each side: y frac{3}{4}x - frac{3}{2} - 3

Combining the constants: y frac{3}{4}x - frac{3}{2} - frac{6}{2}

Reducing the fractions: y frac{3}{4}x - frac{9}{2}

Step 4: Convert to Standard Form (Optional)

To convert the equation y frac{3}{4}x - frac{9}{2} to standard form Ax By C 0, we can multiply through by 4 to eliminate the fraction.

Multiplying by 4: 4y 3x - 18

Rearranging: -3x 4y - 18 0

Thus, the equation of the line parallel to 3x - 4y 6 0 and passing through the point (2, -3) is: 3x - 4y - 18 0.

Perpendicular Lines: A Different Perspective

Perpendicular lines are lines that intersect at right angles. The slopes of perpendicular lines are negative reciprocals of each other. Let's explore how to find the equation of a line perpendicular to a given line and passing through a specific point.

Example of a Perpendicular Line

Given the line 3x 4y - 5 0, we can rewrite it in slope-intercept form: 4y -3x 5 or y -frac{3}{4}x frac{5}{4}. Thus, the slope of the given line is m -frac{3}{4}.

The slope of the line perpendicular to it would be the negative reciprocal, which is frac{4}{3}.

We know the line passes through the point (2, -3), so we can use the point-slope form: y - (-3) frac{4}{3}(x - 2).

Adding the -3: y 3 frac{4}{3}x - frac{8}{3}.

Subtracting 3 from each side: y frac{4}{3}x - frac{8}{3} - 3.

Combining the constants: y frac{4}{3}x - frac{8}{3} - frac{9}{3}.

Reducing: y frac{4}{3}x - frac{17}{3}.

Conclusion

In summary, understanding how to find the equation of a line parallel to a given line and passing through a specific point involves identifying the slope of the given line, using the point-slope form, and converting to standard form if necessary. Similarly, finding the equation of a perpendicular line involves determining the negative reciprocal of the slope and using the same point-slope form.

Related Keywords

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