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What is the angle between the lines y = 2x + 1 and y = -0.5x + 3?

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Question: What is the angle between the lines y = 2x + 1 and y = -0.5x + 3?

Options:

  1. 90 degrees
  2. 60 degrees
  3. 45 degrees
  4. 30 degrees

Correct Answer: 90 degrees

Solution: 

The slopes are m1 = 2 and m2 = -0.5. The angle θ is given by tan(θ) = |(m1 - m2) / (1 + m1*m2)| = |(2 + 0.5) / (1 - 1)|, which is undefined, indicating 90 degrees.

What is the angle between the lines y = 2x + 1 and y = -0.5x + 3?

Practice Questions

Q1
What is the angle between the lines y = 2x + 1 and y = -0.5x + 3?
  1. 90 degrees
  2. 60 degrees
  3. 45 degrees
  4. 30 degrees

Questions & Step-by-Step Solutions

What is the angle between the lines y = 2x + 1 and y = -0.5x + 3?
Correct Answer: 90 degrees
  • Step 1: Identify the equations of the lines. The first line is y = 2x + 1 and the second line is y = -0.5x + 3.
  • Step 2: Find the slope of the first line (m1). The slope is the coefficient of x, so m1 = 2.
  • Step 3: Find the slope of the second line (m2). The slope is the coefficient of x, so m2 = -0.5.
  • Step 4: Use the formula to find the angle θ between the two lines: tan(θ) = |(m1 - m2) / (1 + m1*m2)|.
  • Step 5: Substitute the values of m1 and m2 into the formula: tan(θ) = |(2 - (-0.5)) / (1 + 2 * -0.5)|.
  • Step 6: Simplify the expression: tan(θ) = |(2 + 0.5) / (1 - 1)|.
  • Step 7: Notice that the denominator (1 - 1) equals 0, which means tan(θ) is undefined.
  • Step 8: When tan(θ) is undefined, it indicates that the angle θ is 90 degrees.
  • Slope of a Line – Understanding how to find the slope from the equation of a line in slope-intercept form (y = mx + b).
  • Angle Between Two Lines – Using the formula for the angle between two lines based on their slopes.
  • Undefined Values in Mathematics – Recognizing when a mathematical expression is undefined and its implications.
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