The coordinates for point R will be (-1, -6). This is because a rectangle has opposite sides and as you plot your rectangle with these defines points along with that of R, you will be able to successfully achieve a perfect rectangle.
Answer:
coordinates of R(x,y) is R(6,-2).
Step-by-step explanation:
We have given the coordinates of rectangle PQRS as: P(-1,2), Q(2,4), R(x,y), and S(3,-4).
We know the slope formula:
Slope = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
Now, by applying slope formula
Slope of PQ = [tex]\frac{4-2}{2-(-1)}[/tex]
= [tex]\frac{2}{3}[/tex]
Slope of RS = [tex]\frac{-4-y}{3-x}[/tex]
Slope of QR = [tex]\frac{y-4}{x-2}[/tex]
We know that, slopes of PQ and RS are equal because these are parallel to each other.
Slope of PQ = Slope of RS
[tex]\frac{2}{3}[/tex] = [tex]\frac{-4-y}{3-x}[/tex]
2(3-x) = 3(-4-y)
6-2x = -12-3y
2x-3y = 18...........equation(1)
Slopes of perpendicular lines are negative reciprocal. Therefore,
Slope of PQ = [tex]\frac{-1}{slope \ of \ QR}[/tex]
[tex]\frac{2}{3}[/tex] =[tex]\frac{-1}{\frac{y-4}{x-2} }[/tex]
[tex]\frac{2}{3}[/tex] = [tex]\frac{-x+2}{y-4}[/tex]
On cross multiplication,
2(y-4) = 3(-x+2)
2y-8 = -3x+6
3x+2y = 14.............equation(2)
Now, multiplying equation(1) by 2 and equation(2) by 3 and then equations are adding..
So, 4x-6y=36
9x+6y=42
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13x=78
x=6
Putting the value of x in equation(2)
18+2y=14
2y=-4
y=-2
So, coordinates of R(x,y) is R(6,-2).
