If you are given 3 vertices of a parallelogram, as a Geometry student, you will be asked to find the exact coordinates of the 4th vertex. Â Just so that you can put your mind at ease, there can be different answers depending on how you decide to "draw" or "shape" your parallelogram. Â So if your answer is different than another student, it does not mean that you have done it incorrectly.
Suppose I am given 3 points as such, and I am looking for the coordinates of D:
A (2, 3)
B (8, 1)
C (11, 5)
D (x, y)
The line segments that make up our parallelogram is AD is parallel to BC, and CD is parallel to AB. Â Since the definition of a parallelogram is a quadrilateral with 2 pairs of parallel sides, we can use this fact to find the coordinate to the 4 vertex. Â A pair of parallel side also means that these 2 lines have the same slope. Â So if we do this on both pairs of parallel lines, we should have 2 equations with 2 variables, which should be simple to solve.
Slope of CD = Slope Slope of AB
(y - 5)/(x - 11) = (1 - 3)/(8 - 2)
(y - 5)/(x - 11) = -2/6
6(y - 5) = -2(x - 11)
6y - 30 = -2x + 22
equation #1: Â 6y + 2x = 52
Slope of AD = Slope of BC
(y - 3)/(x - 2) = (5 - 1)/(11 - 8)
(y - 3)/(x - 2) = 4/3
3(y - 3) = 4(x - 2)
3y -9 = 4x - 8
equation #2: Â 3y - 4x = 1
We now we have 2 equations with 2 variables, and we can use any method (substitution, linear combination, matrix) to solve for the x and y coordinates. Â
equation #1: Â 6y + 2x = 52
equation #2: Â 3y - 4x = 1
Multiply equation by -2
Equation 2: -2(3y - 4x = 1)
Equation 2: Â -6y + 8x = -2
Add equation 1 to equation 2
equation 1: Â 6y + 2x = 52
equation 2: Â -6y + 8x = -2
eq 1 + eq 2: Â 10x = 50
x = 5
Use any of the original equation to solve for y:
6y + 2x = 52
6y + 2(5) = 52
6y = 42
y = 7
So the 4th vertex for my first answer will be (5, 7)
If we draw the parallelogram in a different way using the given vertices, we will be looking for a different 4th vertex. Â So in our original question:
A (2, 3)
B (8, 1)
C (11, 5)
D (x, y)
Our line segments in this case will be draw as such:
AC is parallel to BD and CD is parallel to AB.
Slope of AC = slope of BD
(5 - 3)/(11 - 2) = (y - 1)/(x - 8)
2/9 = (y - 1)/(x - 8)
2(x - 8) = 9(y - 1)
2x - 16 = 9y - 9
equation 1: Â 2x - 9y = 7
Slope of CD = slope of AB
(5 - y)/(11 - x) = (3 - 1)/(2 - 8)
(5 - y)/(11 - x) = 2/-6
-6(5 - y) = 2(11 - x)
-30 + 6y = 22 - 2x
equation 2: Â 2x + 6y = 52
Solve the following linear system:
equation 1: Â 2x - 9y = 7
equation 2: Â 2x + 6y = 52
eq 1 - eq2: Â -15y = -45
-15y = -45
y = 3
Substitute y = 3 into any one of the original equation:
equation 1: Â 2x - 9y = 7
2x - 9(3) = 7
2x - 27 = 7
2x = 34
x = 17
So by drawing the parallelogram in a different way, the 4th vertex of this parallelogram is (17, 3).
Step-by-step explanation:
See Answer