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Difficulty: Very hardCoordinate Geometry: Lines, Slopes, and Distance

In the xyxy-plane, line kk passes through the point (1,−2)(1, -2) and is perpendicular to line mm, which is defined by the equation 3x−4y=123x - 4y = 12. Line kk intersects line nn, defined by the equation y=2x+1y = 2x + 1, at point PP. What is the distance between point PP and the point (3.5,3)(3.5, 3)?

Answer: 5

Answer

5
Rewriting the equation of line mm, 3x−4y=123x - 4y = 12, in slope-intercept form gives y=34x−3y = \frac{3}{4}x - 3, so line mm has a slope of 34\frac{3}{4}. Since line kk is perpendicular to line mm, its slope is the negative reciprocal, −43-\frac{4}{3}. Using the point (1,−2)(1, -2), line kk has the equation y=−43x−23y = -\frac{4}{3}x - \frac{2}{3}. Setting this equal to the equation of line nn (y=2x+1y = 2x + 1) yields −43x−23=2x+1-\frac{4}{3}x - \frac{2}{3} = 2x + 1, which solves to x=−0.5x = -0.5 and y=0y = 0, giving the intersection point P(−0.5,0)P(-0.5, 0). Finally, calculating the distance between P(−0.5,0)P(-0.5, 0) and (3.5,3)(3.5, 3) using the distance formula gives (3.5−(−0.5))2+(3−0)2=42+32=25=5\sqrt{(3.5 - (-0.5))^2 + (3 - 0)^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5.

Step-by-Step Solution

1
Determine the slope of line mm
The slope of line mm is 34\frac{3}{4}
Rewriting 3x−4y=123x - 4y = 12 in slope-intercept form gives y=34x−3y = \frac{3}{4}x - 3, so the slope is 34\frac{3}{4}.
2
Determine the slope of line kk
The slope of line kk is −43-\frac{4}{3}
Perpendicular lines have slopes that are negative reciprocals of each other.
3
Write the equation of line kk
The equation of line kk is y=−43x−23y = -\frac{4}{3}x - \frac{2}{3}
Using point-slope form with (1,−2)(1, -2) gives y−(−2)=−43(x−1)y - (-2) = -\frac{4}{3}(x - 1), which simplifies to y=−43x−23y = -\frac{4}{3}x - \frac{2}{3}.
4
Find the coordinates of intersection point PP
Point PP has coordinates (−0.5,0)(-0.5, 0)
Setting −43x−23=2x+1-\frac{4}{3}x - \frac{2}{3} = 2x + 1 yields −103x=53  ⟹  x=−0.5-\frac{10}{3}x = \frac{5}{3} \implies x = -0.5. Substituting x=−0.5x = -0.5 into y=2x+1y = 2x + 1 yields y=0y = 0.
5
Calculate the distance between P(−0.5,0)P(-0.5, 0) and (3.5,3)(3.5, 3)
The distance is 55
Applying the distance formula yields d=(3.5−(−0.5))2+(3−0)2=42+32=25=5d = \sqrt{(3.5 - (-0.5))^2 + (3 - 0)^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5.

Key Concept

Perpendicular line slopes, finding intersection of two lines, and applying the distance formula.
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Topics

Coordinate Geometry: Lines, Slopes, and Distance

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