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Let's solve this step by step.

We are given the equation of line \( r \): 
\[
y - 10 = -2(x - 6)
\]

### Step 1: Convert the equation of line \( r \) to slope-intercept form
We start by expanding and simplifying the given equation to write it in the slope-intercept form \( y = mx + b \).

\[
y - 10 = -2(x - 6)
\]
Distribute the \(-2\):
\[
y - 10 = -2x + 12
\]
Add 10 to both sides:
\[
y = -2x + 22
\]
So, the equation of line \( r \) is \( y = -2x + 22 \), where the slope \( m_r \) of line \( r \) is \(-2\).

### Step 2: Determine the slope of line \( s \)
Line \( s \) is perpendicular to line \( r \). The slope of perpendicular lines is the negative reciprocal of each other. The slope of line \( r \) is \(-2\), so the slope of line \( s \) (denoted \( m_s \)) is:
\[
m_s = \frac{1}{2}
\]

### Step 3: Use the point-slope form for line \( s \)
We are told that line \( s \) passes through the point \( (1, 2) \). Using the point-slope form of a line equation:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m_s = \frac{1}{2} \), \( x_1 = 1 \), and \( y_1 = 2 \):
\[
y - 2 = \frac{1}{2}(x - 1)
\]
Expand the right side:
\[
y - 2 = \frac{1}{2}x - \frac{1}{2}
\]
Add 2 to both sides:
\[
y = \frac{1}{2}x + \frac{3}{2}
\]

### Final Answer:
The equation of line \( s \) in slope-intercept form is:
\[
y = \frac{1}{2}x + \frac{3}{2}
\]

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the equation of a line parallel to line \( r \) that passes through the point \( (3, 5) \)?
2. How do you find the equation of a line given two points?
3. What is the relationship between the slopes of perpendicular lines?
4. How would the equation of line \( s \) change if it passed through the point \( (2, -1) \)?
5. How can you determine if two lines are parallel or perpendicular by their equations?

**Tip:** When dealing with perpendicular lines, always remember that the product of their slopes is \(-1\).

Line r has an equation of y - 10 = -2(x - 6). Line s is perpendicular to line r and passes through (1, 2). What is the equation of line s? Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

Learn how to find the equation of a line perpendicular to y - 10 = -2(x - 6) that passes through (1, 2). This problem demonstrates key algebraic concepts, including the use of slope-intercept and point-slope forms. Suitable for students in Grades 8-10.

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