Understanding Perpendicular Lines and Their Slopes: A Detailed Analysis

In mathematics, the concept of perpendicular lines and their slopes is a fundamental topic in coordinate geometry. Understanding the relationship between the slopes of two perpendicular lines can help in solving various geometric problems. This article provides a detailed explanation of how to find the slope of a line and how to determine if two lines are perpendicular.

Problem Statement

The problem at hand involves two pairs of points, A(-1, y), B(2, -10), C(-3, -7), and D(5, -5). We are given the condition that line AB is perpendicular to line CD. The goal is to find the value of y.

Solution

Method 1: Using Slopes

First, let's find the slope of line AB and line CD using the formula for the slope of a line, which is m (y2 - y1) / (x2 - x1).

Line AB: A(-1, y) and B(2, -10)

The slope of line AB is:

[ m_1 frac{-10 - y}{2 - (-1)} frac{-10 - y}{3} ]

Line CD: C(-3, -7) and D(5, -5)

The slope of line CD is:

[ m_2 frac{-5 - (-7)}{5 - (-3)} frac{2}{8} frac{1}{4} ]

Since line AB is perpendicular to line CD, the product of their slopes should be -1:

[ m_1 cdot m_2 -1 ]

Substituting the values of (m_1) and (m_2), we get:

[ frac{-10 - y}{3} cdot frac{1}{4} -1 ]

Simplifying the equation, we find:

[ frac{-10 - y}{3} -4 ]

Multiplying both sides by 3, we get:

[ -10 - y -12 ]

Adding 12 to both sides, we get:

[ y 2 ]

Hence, the value of y is 2.

Method 2: Using Vector Dot Product

We can also solve the same problem using the vector dot product approach. The vectors for lines AB and CD are:

[ vec{AB} (2 - (-1), -10 - y) (3, -10 - y) ] [ vec{CD} (5 - (-3), -5 - (-7)) (8, 2) ]

The dot product of vectors AB and CD is:

[ vec{AB} cdot vec{CD} 3 cdot 8 (-10 - y) cdot 2 24 - 20 - 2y 4 - 2y ]

Since AB is perpendicular to CD, the dot product is zero:

[ 4 - 2y 0 ]

Solving for y, we get:

[ 2y 4 ]

[ y 2 ]

Therefore, the value of y is 2.

Method 3: Using Negative Reciprocals

An alternative method is to use the fact that the slope of a line perpendicular to another line is the negative reciprocal of the other line's slope.

The slope of line AB is [ frac{-10 - y}{3} ] and the slope of line CD is [ frac{1}{4} ].

Since the lines are perpendicular, the product of their slopes is -1:

[ frac{-10 - y}{3} cdot frac{1}{4} -1 ]

Multiplying both sides by 12, we get:

[ -10 - y -12 ]

[ y 2 ]

Hence, the value of y is 2.

Conclusion

In this article, we have discussed three different methods to find the value of y given that line AB is perpendicular to line CD. These methods—using slopes, vector dot product, and negative reciprocals—highlight the importance of understanding the relationship between the slopes of perpendicular lines.

For more detailed explanations and additional examples, refer to the following resources:

Math Is Fun Khan Academy

If you have any questions or need further clarification, feel free to contact me.

Best Regards,

Mashaim Javaid