What is the equation of a line perpendicular to the line passing through the points (2, 1) and (4, 5)?

To find the equation of a line perpendicular to the line passing through the points \((2, 1)\) and \((4, 5)\), we follow these steps:

### 1. Find the slope of the given line

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the given points \((2, 1)\) and \((4, 5)\):

\[
m = \frac{5 - 1}{4 - 2} = \frac{4}{2} = 2
\]

So, the slope of the given line is \(2\).

### 2. Find the slope of the perpendicular line

The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. If the slope of the original line is \(m\), the slope of the perpendicular line is \(-\frac{1}{m}\).

Given the slope of the original line is \(2\), the slope of the perpendicular line will be:

\[
-\frac{1}{2}
\]

### 3. Use the point-slope form to write the equation

To find the specific equation of the perpendicular line, we would need a point through which it passes. However, since we only need the general equation, we can represent it using the slope-intercept form or point-slope form.

Assuming the line passes through a generic point \((x_1, y_1)\), we use the point-slope form:

\[
y - y_1 = m(x - x_1)
\]

Substituting \(m = -\frac{1}{2}\) and assuming it passes through a generic point \((x_1, y_1)\):

\[
y - y_1 = -\frac{1}{2}(x - x_1)
\]

If no specific point is provided, this equation is a general form for the line perpendicular to the given line.

### 4. For a specific example, assume it passes through the point \((2, 1)\)

If we consider that the line passes through the point \((2, 1)\), then:

\[
y - 1 = -\frac{1}{2}(x - 2)
\]

### 5. Simplify the equation

Expanding and rearranging to the slope-intercept form:

\[
y - 1 = -\frac{1}{2}x + 1
\]

\[
y = -\frac{1}{2}x + 2
\]

Therefore, the equation of the line perpendicular to the line passing through \((2, 1)\) and \((4, 5)\) and passing through the point \((2, 1)\) is:

\[
\boxed{y = -\frac{1}{2}x + 2}
\]

Would you like further details or have any questions?

Here are some related questions for further practice:

1. What is the equation of a line passing through the point \((0, 3)\) and perpendicular to the line \(y = 3x - 1\)?
2. Find the slope of a line perpendicular to the line given by \(y = -\frac{3}{4}x + 6\).
3. How do you determine if two lines are perpendicular by looking at their slopes?
4. If a line has a slope of \( \frac{5}{3} \), what is the slope of a line perpendicular to it?
5. Write the equation of a line parallel to the line \(2x - 3y = 6\) and passing through the point \((3, -2)\).

**Tip:** The slopes of two perpendicular lines multiply to \(-1\). This is a quick check to verify perpendicularity.

Learn how to find the equation of a line perpendicular to the line passing through the points (2, 1) and (4, 5). This problem covers calculating slopes, determining perpendicular slopes using the negative reciprocal rule, and writing equations in point-slope form. Suitable for Grades 7-9 students.

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