Understanding the Concept of Perpendicular Lines

When working with lines in geometry and algebra, the concept of a perpendicular line is often introduced. A line is considered perpendicular to another if the angle formed between them is 90 degrees. This property helps in determining the relationship between slopes of these lines and allows us to find the equation of a line that passes through a given point and is perpendicular to another line.

Identifying the Slope of the Given Line

To begin, we need to identify the slope of the given line. The equation of the line in question is given as:

y frac{1}{2}x - 5.

The slope of the line in the equation form y mx b is represented by the coefficient of x, which in this case is (frac{1}{2}).

Calculating the Slope of the Perpendicular Line

The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. This means that if the original slope is (m), the slope of the perpendicular line will be (-frac{1}{m}).

Given the slope of the original line is (frac{1}{2}), the slope of the perpendicular line will be:

(m_{text{perpendicular}} -frac{1}{frac{1}{2}} -2).

Finding the Equation of the Perpendicular Line

Now that we have the slope of the line that is perpendicular, we can use the point-slope form to find the equation of the line. The point-slope form is:

(y - y_1 m(x - x_1)),

where ((x_1, y_1)) is the point through which the line passes.

The given point is ((-2, -1)), and the slope (m) of the perpendicular line is (-2).

Substituting these values into the point-slope form, we get:

(y - (-1) -2(x - (-2)))

Simplifying this, we obtain:

(y 1 -2(x 2))

(y 1 -2x - 4)

(y -2x - 4 - 1)

(y -2x - 5)

Therefore, the equation of the line passing through the point ((-2, -1)) and perpendicular to the line (y frac{1}{2}x - 5) is:

y -2x - 5

Conclusion

By understanding the concept of slope and the relationship between slopes of perpendicular lines, we can determine the equation of a line that passes through a given point and is perpendicular to another line. This method is crucial in various fields such as engineering, physics, and computer graphics.