Any two lines intersect means perpendicular line? For example, y=2x-5 and y=x-2 intersect at point (3,1), is this perpendicular? But I also notice that both given m or Slope of lines from equation are different and not negative reciprocal of another. Please clear my doubts.

No, not all intersecting lines are perpendicular. For two lines to be perpendicular, the slopes of the lines must be negative reciprocals of each other. This means that if one line has a slope of \( m_1 \), the other line must have a slope of \( m_2 \) such that \( m_1 \cdot m_2 = -1 \).

Let's check the given lines:

1. **Line 1**: \( y = 2x - 5 \)
   - Slope (\( m_1 \)): 2

2. **Line 2**: \( y = x - 2 \)
   - Slope (\( m_2 \)): 1

For these lines to be perpendicular, the product of their slopes should be \(-1\):

\[ m_1 \cdot m_2 = 2 \cdot 1 = 2 \]

Since \( 2 \neq -1 \), these lines are not perpendicular. 

### Intersection Point
The lines intersect at the point (3, 1), which can be verified by solving the system of equations formed by setting the equations of the lines equal to each other:

\[ 2x - 5 = x - 2 \]

Solving for \( x \):

\[ 2x - x = -2 + 5 \]
\[ x = 3 \]

Now substitute \( x = 3 \) into either of the original equations to find \( y \):

\[ y = 2(3) - 5 = 6 - 5 = 1 \]

or

\[ y = 3 - 2 = 1 \]

So the point of intersection is indeed (3, 1), but the lines are not perpendicular.

### Summary
- Two lines are perpendicular if and only if their slopes are negative reciprocals of each other.
- The given lines \( y = 2x - 5 \) and \( y = x - 2 \) intersect but are not perpendicular because their slopes are not negative reciprocals.

Would you like more details or have any questions?

### Questions for you to ask:
1. How do you determine if two lines are parallel?
2. What is the significance of the slope in a linear equation?
3. Can you explain how to find the intersection point of two lines algebraically?
4. What are some applications of perpendicular lines in real life?
5. How does changing the slope affect the angle between two lines?

### Tip:
When dealing with slopes, always remember that the slope-intercept form of a line \( y = mx + b \) can help you quickly identify the slope (\( m \)) and the y-intercept (\( b \)).

Explore the concept of perpendicular lines using the example of y=2x-5 and y=x-2. Learn why these lines intersect but are not perpendicular, based on their slopes and the condition for perpendicularity. Suitable for high school students and anyone interested in understanding line equations and their intersections.

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