https://file.aiquickdraw.com/mathbot/file/17261474071721qi3y17r.jpg

We are given two line segments:  
- Line segment joining \( U(-3, p) \) and \( V(-6, 5) \)
- Line segment joining \( X(4, 2) \) and \( Y(9, 0) \)

These two segments are perpendicular to each other.

### Step 1: Find the slope of line segment \( XY \)
The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

For points \( X(4, 2) \) and \( Y(9, 0) \):

\[
m_{XY} = \frac{0 - 2}{9 - 4} = \frac{-2}{5}
\]

Thus, the slope of line segment \( XY \) is \( m_{XY} = -\frac{2}{5} \).

### Step 2: Find the slope of line segment \( UV \)
For points \( U(-3, p) \) and \( V(-6, 5) \):

\[
m_{UV} = \frac{5 - p}{-6 - (-3)} = \frac{5 - p}{-3}
\]

### Step 3: Use the condition of perpendicularity
For two lines to be perpendicular, the product of their slopes must be equal to \(-1\):

\[
m_{XY} \cdot m_{UV} = -1
\]

Substitute the slopes:

\[
\left(-\frac{2}{5}\right) \cdot \frac{5 - p}{-3} = -1
\]

Simplify the equation:

\[
\frac{2(5 - p)}{15} = -1
\]

Multiply both sides by 15:

\[
2(5 - p) = -15
\]

Expand:

\[
10 - 2p = -15
\]

### Step 4: Solve for \( p \)
Subtract 10 from both sides:

\[
-2p = -25
\]

Divide by -2:

\[
p = \frac{-25}{-2} = 12.5
\]

Thus, the value of \( p \), to the nearest tenth, is \( p = 12.5 \).

Would you like further details or explanations on any step? 

Here are some related questions:
1. How do you find the slope between two points?
2. What is the general condition for two lines to be perpendicular?
3. How can you use the slopes of perpendicular lines in geometry problems?
4. What is the significance of solving for unknown variables in line equations?
5. Can you always find a unique solution for \( p \) in such perpendicularity conditions?

**Tip:** When solving for perpendicular lines, remember that their slopes must multiply to \(-1\).

The line segment joining U(-3, p) and V(-6, 5) is perpendicular to the line segment joining X(4, 2) and Y(9, 0). The value of p, to the nearest tenth is __.

Determine the value of p for the perpendicular line segments joining points U(-3, p) and V(-6, 5) and points X(4, 2) and Y(9, 0). This problem involves understanding slopes, the condition for perpendicularity, and solving for unknown variables. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation