Question

vito is writing a coordinate proof involving a parallelogram. knowing that the opposite sides of a parallelogram are congruent, vito places his parallelogram on the coordinate plane such that bne vertex is at the origin and one side lies along the x - axis. what coordinates should he assign to the fourth vertex of the parallelogram? (2a, 2c) (2c + 2a, 2b) (2c, 2b) (2c + 2b, 2a)

Explanation:

Step1: Use the property of parallelogram

In a parallelogram, if we have three vertices \(A(0,0)\), \(B(2c,0)\), \(C(2a,2b)\), and the fourth vertex \(D(x,y)\). The vector \(\overrightarrow{AB}=\overrightarrow{DC}\).
The vector \(\overrightarrow{AB}=(2c - 0,0 - 0)=(2c,0)\). Let the coordinates of the fourth vertex be \((x,y)\), then \(\overrightarrow{DC}=(2a - x,2b - y)\).

Step2: Solve for coordinates

Since \(\overrightarrow{AB}=\overrightarrow{DC}\), we have the following system of equations:
\(

$$\begin{cases}2c=2a - x\\0 = 2b - y\end{cases}$$

\)
From \(0 = 2b - y\), we get \(y = 2b\). From \(2c=2a - x\), we get \(x=2a + 2c\).

Another way: Using the mid - point formula. The mid - point of the diagonals of a parallelogram are the same. Let the vertices be \(A(0,0)\), \(B(2c,0)\), \(C(2a,2b)\) and \(D(x,y)\).
The mid - point of \(AC\) is \((\frac{0 + 2a}{2},\frac{0+2b}{2})=(a,b)\). The mid - point of \(BD\) is \((\frac{2c + x}{2},\frac{0 + y}{2})\).
Since \(\frac{2c + x}{2}=a\) and \(\frac{y}{2}=b\), we solve \(x=2a + 2c\) and \(y = 2b\)

Answer:

\((2c + 2a,2b)\)