Alright so I just started with math and I am stuck on this question:
The points (-4, 6), (5,7), (6, -2) and (-6, -4) makes a quadrangel.
Determine the coordinates for the exact point where the diagonals of the quadrangel cut eachother.
I've translated from Swedish so I hope you guys can understand it!
This is what I've done so far.
I started by drawing out line A (-4, 6) and (6, -2)
The k in line A is: \(k = \Delta Y / \Delta X\)\(\Delta Y is 8\)\(\Delta X is 10\)
Meaning that k = 8/10 = 0.8
The "m" in line a is: y = 0.8 * x + m
I'll input the first coordinates: 6 = 0.8 * -4 = m
6 = -3.2 + m
6 = -3.2 + 9.2
Now I've calculated that K is 0.8 and m is 9.2
Where do i go from here?
+26400 The points (-4, 6), (5,7), (6, -2) and (-6, -4) makes a quadrangel. {nl} Determine the coordinates for the exact point where the diagonals of the quadrangel cut eachother.
In General: cut two lines.
Line 1 with two Points: \(\dbinom{x_1}{y_1}\) and \(\dbinom{x_2}{y_2}\)
Line 2 with two Points: \(\dbinom{x_3}{y_3}\) and \( \dbinom{x_4}{y_4}\)
The coordinates for the cut eachother is \(\dbinom{x_c}{y_c}\)
\(\boxed{~ \begin{array}{lcl} x_c &=& x_3 + k\cdot (x_4-x_3)\\ y_c &=& y_3 + k\cdot (y_4-y_3)\\ k &=& \dfrac{ (x_3-x_1)(y_2-y_1)-(y_3-y_1)(x_2-x_1) }{(x_2-x_1)(y_4-y_3)-(x_4-x_3)(y_2-y_1)} \end{array} ~}\)
Calculation:
\(\begin{array}{lcl} \text{Line 1 with two Points:} \dbinom{x_1=-6}{y_1=-4} \text{ and } \dbinom{x_2=5}{y_2=7}\\\\ \text{Line 2 with two Points:} \dbinom{x_3=6}{y_3=-2} \text{ and } \dbinom{x_4=-4}{y_4=6}\\\\ \end{array}\\ \begin{array}{lcl} k &=& \dfrac{ [6-(-6)][7-(-4)]-[-2-(-4)][5-(-6)] } { [5-(-6)][6-(-2)]-[-4-(-6)][7-(-4)] }\\\\ k &=& \dfrac{ 12\cdot 11 - 2\cdot 11 } { 11\cdot 8 + 10\cdot 11 }\\\\ k &=& \dfrac{ 12 - 2 } { 8 + 10 }\\\\ k &=& \dfrac{ 10 } { 18 }\\\\ k &=& \dfrac{ 5 } { 9 }\\\\ x_c &=& 6 + \dfrac{ 5 } { 9 }\cdot (-4-6)\\ x_c &=& 6 - \dfrac{ 5\cdot 10 } { 9 }\\ x_c &=& \dfrac{ 54-50 } { 9 }\\ \mathbf{ x_c }& \mathbf{=} & \mathbf{ \dfrac{ 4 } { 9 } }\\\\ y_c &=& -2 + \dfrac{ 5 } { 9 }\cdot [6-(-2)]\\ y_c &=& -2 + \dfrac{ 5\cdot 8 } { 9 }\\ y_c &=& \dfrac{ -18+40 } { 9 }\\ \mathbf{ y_c } & \mathbf{=} & \mathbf{ \dfrac{ 22 } { 9 } } \end{array}\)
+118724 I have put our question in the eye of the other answerers.
I hoipe that you get your answer :)
http://web2.0calc.com/questions/unanswered-questions_26956
+26400 The points (-4, 6), (5,7), (6, -2) and (-6, -4) makes a quadrangel. {nl} Determine the coordinates for the exact point where the diagonals of the quadrangel cut eachother.
In General: cut two lines.
Line 1 with two Points: \(\dbinom{x_1}{y_1}\) and \(\dbinom{x_2}{y_2}\)
Line 2 with two Points: \(\dbinom{x_3}{y_3}\) and \( \dbinom{x_4}{y_4}\)
The coordinates for the cut eachother is \(\dbinom{x_c}{y_c}\)
\(\boxed{~ \begin{array}{lcl} x_c &=& x_3 + k\cdot (x_4-x_3)\\ y_c &=& y_3 + k\cdot (y_4-y_3)\\ k &=& \dfrac{ (x_3-x_1)(y_2-y_1)-(y_3-y_1)(x_2-x_1) }{(x_2-x_1)(y_4-y_3)-(x_4-x_3)(y_2-y_1)} \end{array} ~}\)
Calculation:
\(\begin{array}{lcl} \text{Line 1 with two Points:} \dbinom{x_1=-6}{y_1=-4} \text{ and } \dbinom{x_2=5}{y_2=7}\\\\ \text{Line 2 with two Points:} \dbinom{x_3=6}{y_3=-2} \text{ and } \dbinom{x_4=-4}{y_4=6}\\\\ \end{array}\\ \begin{array}{lcl} k &=& \dfrac{ [6-(-6)][7-(-4)]-[-2-(-4)][5-(-6)] } { [5-(-6)][6-(-2)]-[-4-(-6)][7-(-4)] }\\\\ k &=& \dfrac{ 12\cdot 11 - 2\cdot 11 } { 11\cdot 8 + 10\cdot 11 }\\\\ k &=& \dfrac{ 12 - 2 } { 8 + 10 }\\\\ k &=& \dfrac{ 10 } { 18 }\\\\ k &=& \dfrac{ 5 } { 9 }\\\\ x_c &=& 6 + \dfrac{ 5 } { 9 }\cdot (-4-6)\\ x_c &=& 6 - \dfrac{ 5\cdot 10 } { 9 }\\ x_c &=& \dfrac{ 54-50 } { 9 }\\ \mathbf{ x_c }& \mathbf{=} & \mathbf{ \dfrac{ 4 } { 9 } }\\\\ y_c &=& -2 + \dfrac{ 5 } { 9 }\cdot [6-(-2)]\\ y_c &=& -2 + \dfrac{ 5\cdot 8 } { 9 }\\ y_c &=& \dfrac{ -18+40 } { 9 }\\ \mathbf{ y_c } & \mathbf{=} & \mathbf{ \dfrac{ 22 } { 9 } } \end{array}\)
