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 th equation of line s

We are given the equation of line \( r \) in point-slope form:

\[
y + 10 = -2(x - 6)
\]

We need to find the equation of line \( s \), which is perpendicular to line \( r \) and passes through the point \( (1, 2) \).

### Step 1: Find the slope of line \( r \)

In the point-slope form of the equation of line \( r \):
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope. From the equation \( y + 10 = -2(x - 6) \), we can see that the slope of line \( r \) is \( m_r = -2 \).

### Step 2: Find the slope of line \( s \)

Since line \( s \) is perpendicular to line \( r \), the slope of line \( s \) will be the negative reciprocal of the slope of line \( r \). The negative reciprocal of \( -2 \) is:

\[
m_s = \frac{1}{2}
\]

### Step 3: Use the point-slope form for line \( s \)

Now that we have the slope of line \( s \), \( m_s = \frac{1}{2} \), and we know it passes through the point \( (1, 2) \), we can use the point-slope form to write the equation of line \( s \):
\[
y - y_1 = m_s(x - x_1)
\]
Substituting \( m_s = \frac{1}{2} \) and the point \( (1, 2) \):
\[
y - 2 = \frac{1}{2}(x - 1)
\]

### Step 4: Simplify the equation

To get the equation into slope-intercept form, simplify:
\[
y - 2 = \frac{1}{2}x - \frac{1}{2}
\]
Add 2 to both sides:
\[
y = \frac{1}{2}x - \frac{1}{2} + 2
\]
\[
y = \frac{1}{2}x + \frac{3}{2}
\]

Thus, the equation of line \( s \) is:

\[
y = \frac{1}{2}x + \frac{3}{2}
\]

Would you like more details or clarifications on any of the steps?

Here are 5 related questions you could explore:
1. How do you find the perpendicular slope of a line?
2. What is the significance of the point-slope form of a line?
3. How can you convert point-slope form to slope-intercept form?
4. What does it mean geometrically for two lines to be perpendicular?
5. How can you find the equation of a line that passes through two points?

**Tip**: To find the negative reciprocal of a fraction \( \frac{a}{b} \), simply flip the fraction and change the sign: \( \frac{a}{b} \rightarrow -\frac{b}{a} \).

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

Find the equation of a line perpendicular to y + 10 = -2(x - 6) that passes through the point (1, 2). This algebra problem involves point-slope form, slope-intercept form, and the concept of perpendicular slopes. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation