Lines $a$ and $b$ are perpendicular. If the slope of line $a$ is 3, what is the slope of line $b$?
A. -3
B. 3
C. $-\frac{1}{3}$
D. $\frac{1}{3}$
Answer (1)
The problem states that lines a and b are perpendicular and the slope of line a is 3.
Recall that the product of the slopes of perpendicular lines is -1.
Substitute the given slope into the equation m a m b = − 1 .
Solve for m b to find the slope of line b : − 3 1 .
Explanation
Problem Analysis We are given two lines, a and b , that are perpendicular. The slope of line a is 3. We need to find the slope of line b .
Key Concept: Perpendicular Lines The key concept here is that the product of the slopes of two perpendicular lines is -1. Let m a be the slope of line a and m b be the slope of line b . Then, we have:
m a m b = − 1
Substitute the given slope We are given that m a = 3 . Substituting this into the equation, we get:
3 m b = − 1
Solve for the unknown slope Now, we solve for m b by dividing both sides of the equation by 3:
m b = 3 − 1 = − 3 1
Final Answer Therefore, the slope of line b is − 3 1 .
Examples
Understanding perpendicular slopes is crucial in many real-world applications, such as designing roads and buildings. For example, when building a ramp that needs to be perpendicular to a wall, knowing the slope of the wall allows you to calculate the exact slope needed for the ramp to ensure it meets the wall at a 90-degree angle. This ensures structural integrity and safety in construction projects.
