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Difficulty: HardParallel and Perpendicular Lines

In the standard (x,y)(x, y) coordinate plane, one diagonal of a square lies along the line with equation y=3x−4y = 3x - 4. If one of the vertices of the square that does not lie on this diagonal is located at the point (2,8)(2, 8), which of the following equations represents the line containing the other diagonal of the square?

  1. A
    3x−y=−23x - y = -2
  2. B
    3x+y=143x + y = 14
  3. x+3y=26x + 3y = 26Answer
  4. D
    x−3y=−22x - 3y = -22
  5. E
    x+3y=40x + 3y = 40

Answer

The equation of the line containing the other diagonal is x+3y=26x + 3y = 26.
The correct equation is x+3y=26x + 3y = 26. Since the diagonals of a square are perpendicular, their slopes must be negative reciprocals of each other. The given diagonal has a slope of 33, meaning the other diagonal has a slope of −13-\frac{1}{3}. Since the vertex (2,8)(2, 8) does not satisfy the equation of the first diagonal, it must lie on the second diagonal. Substituting this point into the slope-intercept or point-slope equation yields the standard form equation x+3y=26x + 3y = 26.

Step-by-Step Solution

1
Identify the slope of the given diagonal line.
The slope of the line y=3x−4y = 3x - 4 is 33.
The equation is in slope-intercept form y=mx+by = mx + b, where mm is the slope.
2
Determine the relationship between the two diagonals of a square.
The diagonals of a square are perpendicular to each other.
By geometric definition, the diagonals of any square intersect at right angles.
3
Calculate the slope of the perpendicular diagonal.
The slope of the perpendicular diagonal is −13-\frac{1}{3}.
Perpendicular lines have slopes that are negative reciprocals of each other, so the perpendicular slope is −13-\frac{1}{3}.
4
Identify the point that the perpendicular diagonal passes through.
The perpendicular diagonal passes through the vertex (2,8)(2, 8).
The vertex (2,8)(2, 8) does not lie on the first diagonal because 8≠3(2)−48 \neq 3(2) - 4. Since a square only has two diagonals, any vertex not on the first diagonal must lie on the second diagonal.
5
Find the equation of the perpendicular diagonal using point-slope form.
The equation is x+3y=26x + 3y = 26.
Using the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) with m=−13m = -\frac{1}{3} and point (2,8)(2, 8) yields y−8=−13(x−2)y - 8 = -\frac{1}{3}(x - 2), which simplifies to 3(y−8)=−(x−2)⇒3y−24=−x+2⇒x+3y=263(y - 8) = -(x - 2) \Rightarrow 3y - 24 = -x + 2 \Rightarrow x + 3y = 26.

Key Concept

Diagonals of a square are perpendicular, and perpendicular lines have slopes that are negative reciprocals.
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Topics

Parallel and Perpendicular Lines

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