The line L1 has equation 2x + y = 8. The line L2 passes through the point A (7,4) and is perpendicular to L1. Given that the line L1 and L2 intersect at the point B, find the length of AB

To find the length of \( AB \), we need to first determine the coordinates of the point \( B \) where the lines \( L1 \) and \( L2 \) intersect, and then calculate the distance between points \( A \) and \( B \).

### Step 1: Find the slope of line \( L1 \)
The equation of line \( L1 \) is given by:
\[
2x + y = 8
\]
We can rewrite it in slope-intercept form (\( y = mx + c \)) to find the slope \( m \):
\[
y = -2x + 8
\]
So, the slope of \( L1 \) is \( m_1 = -2 \).

### Step 2: Determine the slope of line \( L2 \)
Since line \( L2 \) is perpendicular to \( L1 \), the slope \( m_2 \) of \( L2 \) is the negative reciprocal of \( m_1 \). Therefore,
\[
m_2 = \frac{1}{2}
\]

### Step 3: Find the equation of line \( L2 \)
Line \( L2 \) passes through point \( A(7, 4) \) and has a slope of \( \frac{1}{2} \). Using the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
Substituting the given point and slope:
\[
y - 4 = \frac{1}{2}(x - 7)
\]
Simplifying, we get:
\[
y = \frac{1}{2}x + \left(4 - \frac{7}{2}\right)
\]
\[
y = \frac{1}{2}x + \frac{1}{2}
\]
So the equation of \( L2 \) is:
\[
y = \frac{1}{2}x + \frac{1}{2}
\]

### Step 4: Find the coordinates of point \( B \)
To find the intersection of \( L1 \) and \( L2 \), we need to solve the system of equations:
1. \( 2x + y = 8 \)
2. \( y = \frac{1}{2}x + \frac{1}{2} \)

Substitute the expression for \( y \) from the second equation into the first equation:
\[
2x + \left(\frac{1}{2}x + \frac{1}{2}\right) = 8
\]
Multiply through by 2 to eliminate the fraction:
\[
4x + x + 1 = 16
\]
Simplify and solve for \( x \):
\[
5x = 15 \quad \Rightarrow \quad x = 3
\]
Now, substitute \( x = 3 \) back into the equation for \( L2 \):
\[
y = \frac{1}{2}(3) + \frac{1}{2} = \frac{3}{2} + \frac{1}{2} = 2
\]
So the coordinates of point \( B \) are \( (3, 2) \).

### Step 5: Calculate the length of \( AB \)
Now that we have the coordinates of \( A(7, 4) \) and \( B(3, 2) \), we can use the distance formula:
\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the coordinates:
\[
AB = \sqrt{(3 - 7)^2 + (2 - 4)^2} = \sqrt{(-4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}
\]

Thus, the length of \( AB \) is \( 2\sqrt{5} \).

Would you like more details or have any questions about this solution?

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Here are five related questions:
1. How do you determine the slope of a line given its equation?
2. What is the significance of the negative reciprocal in finding perpendicular slopes?
3. Can you explain how to convert a linear equation to slope-intercept form?
4. How can you find the distance between two points in a coordinate plane?
5. What are some real-life applications of perpendicular lines?

**Tip:** Remember that two lines are perpendicular if the product of their slopes is \(-1\).

Learn how to find the length of AB where two lines intersect perpendicularly. This problem involves coordinate geometry, lines and slopes, and the distance formula. Suitable for Grades 9-10.

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