The line segment drawn from P(x, 3) to (4, 1) is perpendicular to the segment drawn from (-5, -6) to (4, 1). Find the value of x.


To find the value of \( x \), we can use the concept of perpendicular slopes. The slopes of two perpendicular lines are negative reciprocals of each other.

### Step 1: Find the slope of the segment from \( (-5, -6) \) to \( (4, 1) \)

The slope \( m_1 \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[
m_1 = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substitute the points \( (-5, -6) \) and \( (4, 1) \):

\[
m_1 = \frac{1 - (-6)}{4 - (-5)} = \frac{1 + 6}{4 + 5} = \frac{7}{9}
\]

So, the slope of the first segment is \( \frac{7}{9} \).

### Step 2: Find the slope of the segment from \( P(x, 3) \) to \( (4, 1) \)

Let the slope of this second segment be \( m_2 \). Using the slope formula, we have:

\[
m_2 = \frac{1 - 3}{4 - x} = \frac{-2}{4 - x}
\]

### Step 3: Use the condition of perpendicularity

Since the two line segments are perpendicular, the product of their slopes must be \( -1 \). Therefore:

\[
m_1 \cdot m_2 = -1
\]

Substitute the values of \( m_1 = \frac{7}{9} \) and \( m_2 = \frac{-2}{4 - x} \):

\[
\frac{7}{9} \cdot \frac{-2}{4 - x} = -1
\]

### Step 4: Solve for \( x \)

Multiply both sides of the equation:

\[
\frac{-14}{9(4 - x)} = -1
\]

Now, multiply both sides by \( 9(4 - x) \) to eliminate the denominator:

\[
-14 = -9(4 - x)
\]

Simplify:

\[
-14 = -36 + 9x
\]

Add 36 to both sides:

\[
22 = 9x
\]

Finally, divide by 9:

\[
x = \frac{22}{9}
\]

### Final Answer:
\[
x = \frac{22}{9}
\]

Would you like further details on any step or have any questions?

Here are some related questions to explore:
1. How do you find the midpoint of a line segment between two points?
2. What is the general form of the equation of a line given two points?
3. How do you calculate the distance between two points in the coordinate plane?
4. Can you explain why the product of the slopes of perpendicular lines is always \( -1 \)?
5. What are the properties of parallel and perpendicular lines in geometry?

**Tip:** Always check your slope calculations carefully, as small sign errors can change the final result significantly.

The line segment drawn from P(x, 3) to (4, 1) is perpendicular to the segment drawn from (-5, -6) to (4, 1). Find the value of x.

Explore how to find the value of x when the line segment from P(x, 3) to (4, 1) is perpendicular to the segment from (-5, -6) to (4, 1). This problem involves key concepts in geometry, including slopes and the perpendicularity condition. Suitable for students in Grades 9-11.

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