Write an equation of a line that passes through $(1.4,-3)$ and is perpendicular to the line that passes through $(-3,4)$ and $(8,5)$.
Answer (1)
Calculate the slope of the line passing through ( − 3 , 4 ) and ( 8 , 5 ) as m 1 = 11 1 .
Determine the slope of the perpendicular line as m 2 = − 11 .
Use the point-slope form with the point ( 1.4 , − 3 ) and slope m 2 = − 11 to get y + 3 = − 11 ( x − 1.4 ) .
Simplify the equation to slope-intercept form: y = − 11 x + 12.4 . The final answer is y = − 11 x + 12.4 .
Explanation
Problem Analysis The problem asks us to find the equation of a line that passes through a given point and is perpendicular to another line defined by two points. We will first find the slope of the second line, then determine the slope of the perpendicular line. Finally, we will use the point-slope form to find the equation of the desired line.
Finding the Slope of the First Line Let's find the slope of the line passing through the points ( − 3 , 4 ) and ( 8 , 5 ) . The slope, m 1 , is given by: m 1 = x 2 − x 1 y 2 − y 1 = 8 − ( − 3 ) 5 − 4 = 11 1 So, the slope of the first line is 11 1 .
Finding the Slope of the Perpendicular Line Now, we need to find the slope of the line perpendicular to the first line. If two lines are perpendicular, the product of their slopes is − 1 . Let the slope of the perpendicular line be m 2 . Then: m 1 ⋅ m 2 = − 1 m 2 = − m 1 1 = − 11 1 1 = − 11 So, the slope of the perpendicular line is − 11 .
Using the Point-Slope Form We are given that the line passes through the point ( 1.4 , − 3 ) . We can use the point-slope form of a line, which is: y − y 1 = m ( x − x 1 ) where ( x 1 , y 1 ) is the given point and m is the slope. Substituting the values, we get: y − ( − 3 ) = − 11 ( x − 1.4 ) y + 3 = − 11 x + 15.4
Simplifying to Slope-Intercept Form Now, let's simplify the equation to the slope-intercept form, y = m x + b :
y = − 11 x + 15.4 − 3 y = − 11 x + 12.4 So, the equation of the line is y = − 11 x + 12.4 .
Final Answer The equation of the line that passes through ( 1.4 , − 3 ) and is perpendicular to the line that passes through ( − 3 , 4 ) and ( 8 , 5 ) is: y = − 11 x + 12.4
Examples
Understanding perpendicular lines is crucial in many real-world applications, such as architecture and construction. For example, when designing a building, ensuring that walls are perpendicular to the ground is essential for stability. Similarly, in navigation, understanding perpendicular relationships helps in determining the shortest distance between two points or in creating accurate maps. This problem demonstrates how to calculate the equation of a line that is perpendicular to another, a fundamental concept in ensuring structural integrity and accurate spatial representation.
