Find the perpendicular line of y = -1/2x - 1 that passes through the point (4, 5)
Understand the Problem
The question is asking to find the equation of a line that is perpendicular to the given line y = -1/2x - 1 and passes through the point (4, 5). To solve this, we need to first determine the slope of the given line, then find the negative reciprocal of that slope to get the perpendicular slope, and finally use the point-slope form of a line to write the equation.
Answer
The equation of the line is $y = 2x - 3$.
Answer for screen readers
The equation of the line that is perpendicular to the given line and passes through the point (4, 5) is:
$$ y = 2x - 3 $$
Steps to Solve
- Identify the slope of the given line
The given line is represented by the equation $y = -\frac{1}{2}x - 1$. The coefficient of $x$ in this equation is the slope. Thus, the slope $m_1$ of the given line is $-\frac{1}{2}$.
- Find the negative reciprocal of the slope
To find the slope of the line that is perpendicular to the given line, we take the negative reciprocal of the slope we identified. The negative reciprocal of $-\frac{1}{2}$ is:
$$ m_2 = -\left(-\frac{1}{2}\right)^{-1} = 2 $$
- Use the point-slope formula
We will use the point-slope form of a line, which is given by the formula:
$$ y - y_1 = m(x - x_1) $$
Here, $(x_1, y_1)$ is the point we are given: $(4, 5)$. Inserting the slope $m_2 = 2$, we get:
$$ y - 5 = 2(x - 4) $$
- Simplify the equation
Now we simplify the equation:
$$ y - 5 = 2x - 8 $$
Adding 5 to both sides gives:
$$ y = 2x - 3 $$
The equation of the line that is perpendicular to the given line and passes through the point (4, 5) is:
$$ y = 2x - 3 $$
More Information
This line has a slope of 2, which is perpendicular to the original slope of $-\frac{1}{2}$. The concept of perpendicular lines in geometry states that their slopes multiply to $-1$. This approach is commonly used in analytic geometry.
Tips
- Incorrectly calculating the negative reciprocal: Students might mistakenly forget to take the negative sign, or might not recognize how to convert the fraction properly.
- Mistakes in point-slope form: Ensure that when substituting values into the point-slope formula, both coordinates of the point are included accurately.
