Answer:
Explanation:
To construct a square with one side on the line l, we need to find the equation of a line perpendicular to l that passes through P. The slope of the line l is 3/4, so the slope of any line perpendicular to l is -4/3. Thus, the equation of the line through P with slope -4/3 is
y - 1 = (-4/3)(x + 6)
Simplifying, we get
y = (-4/3)x - 7
To find the intersection of this line with l, we can substitute y = (-4/3)x - 7 into the equation for l:
3x - 4((-4/3)x - 7) = 18
Solving for x, we get x = -24/5. Substituting back, we get y = -13/5.
So the intersection point of the two lines is (-24/5, -13/5), which we'll call Q.
To construct a square with side PQ, we need to find the midpoint M of PQ, and then find the points R and S that are equidistant from M and P.
The midpoint of PQ is
(((-24/5) + (-6))/2, ((-13/5) + 1)/2) = (-15/5, -6/5) = (-3, -1/5)
The distance between P and M is
sqrt((-3 - (-6))^2 + (-1/5 - 1)^2) = sqrt(45.04) ≈ 6.71
So the distance between M and R (or S) is also 6.71.
Since P has coordinates (-6, 1), and the distance from M to P is 6.71, we know that R and S must have coordinates (-6 ± 6.71, 1 ± 6.71), which simplifies to (-12.71, 7.71) and (0.71, -5.71), respectively.
So one possible set of vertices for the square is PQRSA, where P = (-6, 1), Q = (-24/5, -13/5), R = (-12.71, 7.71), S = (0.71, -5.71).