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Difficulty: MediumCoordinate Geometry and Lines

In the xyxy-plane, line kk passes through the point (3,5)(3, 5) and has a slope of −3-3. Line mm is perpendicular to line kk and also passes through the point (3,5)(3, 5). What is the xx-intercept of line mm?

  1. −12-12Answer
  2. B
    44
  3. C
    1818
  4. D
    143\frac{14}{3}
  5. E
    43\frac{4}{3}

Answer

−12-12
The slope of line kk is −3-3, so the slope of perpendicular line mm is its negative reciprocal, 13\frac{1}{3}. Substituting slope 13\frac{1}{3} and point (3,5)(3, 5) into point-slope form gives y−5=13(x−3)y - 5 = \frac{1}{3}(x - 3), which simplifies to y=13x+4y = \frac{1}{3}x + 4. Setting y=0y = 0 to find the xx-intercept gives 0=13x+40 = \frac{1}{3}x + 4, yielding x=−12x = -12.

Step-by-Step Solution

1
Determine the slope of line mm
Since line mm is perpendicular to line kk, its slope is the negative reciprocal of −3-3, which is 13\frac{1}{3}.
Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.
2
Find the equation of line mm
Using point-slope form with point (3,5)(3, 5) and slope 13\frac{1}{3}: y−5=13(x−3)  ⟹  y=13x+4y - 5 = \frac{1}{3}(x - 3) \implies y = \frac{1}{3}x + 4.
A line with slope mm passing through (x1,y1)(x_1, y_1) follows y−y1=m(x−x1)y - y_1 = m(x - x_1).
3
Calculate the xx-intercept of line mm
Set y=0y = 0: 0=13x+4  ⟹  13x=−4  ⟹  x=−120 = \frac{1}{3}x + 4 \implies \frac{1}{3}x = -4 \implies x = -12.
The xx-intercept is the xx-coordinate where the line intersects the xx-axis (y=0y = 0).

Key Concept

Perpendicular Slopes and Line Intercepts
Estimated Time:1m 30s
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Topics

Coordinate Geometry and Lines

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