three vertices of a parallelogram are shown in the figure below. give the coordinates of the fourth vertex.
three vertices of a parallelogram are shown in the figure below. give the coordinates of the fourth vertex.
Answer
Explanation:
Step1: Recall parallelogram property
In a parallelogram, the diagonals bisect each other. Let the vertices be (A(0,0)), (B(- 3,4)), (C(2,6)) and the fourth - vertex be (D(x,y)). The mid - point of the diagonal (AC) is the same as the mid - point of the diagonal (BD). The mid - point formula between two points ((x_1,y_1)) and ((x_2,y_2)) is ((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})). The mid - point of (AC) with (A(0,0)) and (C(2,6)) is ((\frac{0 + 2}{2},\frac{0+6}{2})=(1,3)). The mid - point of (BD) with (B(-3,4)) and (D(x,y)) is ((\frac{-3 + x}{2},\frac{4 + y}{2})).
Step2: Set up equations
Since the mid - points of the diagonals are equal, we have the following two equations: (\frac{-3 + x}{2}=1) and (\frac{4 + y}{2}=3). Solve the first equation (\frac{-3 + x}{2}=1) for (x): Multiply both sides by 2: (-3+x = 2). Add 3 to both sides: (x=2 + 3=5). Solve the second equation (\frac{4 + y}{2}=3) for (y): Multiply both sides by 2: (4 + y=6). Subtract 4 from both sides: (y=6 - 4 = 2).
Answer:
((5,2))