Examkin

Question

Difficulty: MediumParallel and Perpendicular Lines

On a coordinate map of an airport, Runway A is represented by the line 4x−3y=124x - 3y = 12. Runway B is designed to be perpendicular to Runway A, and its path crosses the yy-axis at (0,7)(0, 7). If Runway B passes through a guidance beacon located at the coordinate (p,1)(p, 1), what is the value of pp?

  1. A
    −8-8
  2. B
    −92-\frac{9}{2}
  3. C
    −323-\frac{32}{3}
  4. 88Answer
  5. E
    92\frac{9}{2}

Answer

8
The correct answer is 88. First, find the slope of Runway A by rewriting 4x−3y=124x - 3y = 12 as y=43x−4y = \frac{4}{3}x - 4, which gives a slope of 43\frac{4}{3}. The slope of Runway B must be the negative reciprocal, −34-\frac{3}{4}. Using the given yy-intercept of 77, the equation for Runway B is y=−34x+7y = -\frac{3}{4}x + 7. Substituting the point (p,1)(p, 1) gives 1=−34p+71 = -\frac{3}{4}p + 7. Subtracting 77 from both sides yields −6=−34p-6 = -\frac{3}{4}p. Multiplying both sides by −43-\frac{4}{3} gives p=8p = 8.

Step-by-Step Solution

1
Convert the equation of Runway A, 4x−3y=124x - 3y = 12, into slope-intercept form (y=mx+by = mx + b) to find its slope.
The equation becomes y=43x−4y = \frac{4}{3}x - 4, which shows the slope of Runway A is m1=43m_1 = \frac{4}{3}.
Finding the slope of the first line is necessary to determine the slope of any perpendicular line.
2
Calculate the perpendicular slope (m2m_2) for Runway B by taking the negative reciprocal of the slope of Runway A (m1m_1).
The perpendicular slope is m2=−1m1=−34m_2 = -\frac{1}{m_1} = -\frac{3}{4}.
Perpendicular lines in a coordinate plane have slopes that are negative reciprocals of each other.
3
Write the equation of Runway B using its slope m2=−34m_2 = -\frac{3}{4} and its yy-intercept of 77.
The equation of Runway B is y=−34x+7y = -\frac{3}{4}x + 7.
The slope-intercept form of a line is y=mx+by = mx + b, where mm is the slope and bb is the yy-intercept.
4
Substitute the point (p,1)(p, 1) into the equation of Runway B and solve for pp.
1=−34p+7  ⟹  −6=−34p  ⟹  24=3p  ⟹  p=81 = -\frac{3}{4}p + 7 \implies -6 = -\frac{3}{4}p \implies 24 = 3p \implies p = 8.
Since the guidance beacon lies on Runway B, its coordinates must satisfy the equation of the line.

Key Concept

Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product is −1-1. Once the perpendicular slope and y-intercept are known, the line's equation can be written and solved for a missing coordinate.
Estimated Time:1m 0s
Rate this question

Topics

Parallel and Perpendicular Lines

More questions from this topic