Coordinates

Determine the coordinates of the vertices and the content of the parallelogram, the two sides of which lie on the lines 8x + 3y + 1 = 0, 2x + y-1 = 0 and the diagonal on the line 3x + 2y + 3 = 0

Correct result:

x0 =  -2
y0 =  5
x1 =  1
y1 =  -3
x2 =  5
y2 =  -9
x3 =  8
y3 =  -17
S =  14

Solution:


8x0+3y0+1=0
2x0+y0-1=0

8•x0+3•y0+1=0
2•x0+y0-1=0

8x0+3y0 = -1
2x0+y0 = 1

x0 = -2
y0 = 5

Calculated by our linear equations calculator.

8x1+3y1+1=0
3x1+ 2y1+3=0

8•x1+3•y1+1=0
3•x1+ 2•y1+3=0

8x1+3y1 = -1
3x1+2y1 = -3

x1 = 1
y1 = -3

Calculated by our linear equations calculator.

2x2+y2-1=0
3x2+ 2y2+3=0

2•x2+y2-1=0
3•x2+ 2•y2+3=0

2x2+y2 = 1
3x2+2y2 = -3

x2 = 5
y2 = -9

Calculated by our linear equations calculator.
x=x2+x12=5+12=3 y=y2+y12=(9)+(3)2=6  x3=x0+2 x=(2)+2 3=8x=\dfrac{ x_{2}+x_{1} }{ 2 }=\dfrac{ 5+1 }{ 2 }=3 \ \\ y=\dfrac{ y_{2}+y_{1} }{ 2 }=\dfrac{ (-9)+(-3) }{ 2 }=-6 \ \\ \ \\ x_{3}=-x_{0}+2 \cdot \ x=-(-2)+2 \cdot \ 3=8
y3=y0+2 y=5+2 (6)=17y_{3}=-y_{0}+2 \cdot \ y=-5+2 \cdot \ (-6)=-17
a=(x0x1)2+(y0y1)2=((2)1)2+(5(3))2738.544 h=p,C h=8 x2+3 y2+182+32=8 5+3 (9)+182+321.6386 S=a h=8.544 1.6386=14a=\sqrt{ (x_{0}-x_{1})^2+(y_{0}-y_{1})^2 }=\sqrt{ ((-2)-1)^2+(5-(-3))^2 } \doteq \sqrt{ 73 } \doteq 8.544 \ \\ h=|p,C| \ \\ h=\dfrac{ 8 \cdot \ x_{2}+3 \cdot \ y_{2}+1 }{ \sqrt{ 8^2+3^2 } }=\dfrac{ 8 \cdot \ 5+3 \cdot \ (-9)+1 }{ \sqrt{ 8^2+3^2 } } \doteq 1.6386 \ \\ S=a \cdot \ h=8.544 \cdot \ 1.6386=14



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For Basic calculations in analytic geometry is helpful line slope calculator. From coordinates of two points in the plane it calculate slope, normal and parametric line equation(s), slope, directional angle, direction vector, the length of segment, intersections the coordinate axes etc.
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See also our trigonometric triangle calculator.

 
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