Question: Parallelogram \( DEFG \) is shown below. Give the coordinates of \( F \). \[…

Parallelogram \( DEFG \) is shown below. Give the coordinates of \( F \).

\[ \begin{array}{c} \text{Parallelogram } DEFG \text{ is shown below. Give the coordinates of } F. \\ \begin{tikzpicture} \draw[->] (-1, 0) — (5, 0) node[right] {$x$}; \draw[->] (0, -1) — (0, 5) node[above] {$y$}; \draw[thick, blue] (0, 0) — (4, 0) — (4, 3) — (0, 3) — cycle; \draw[thick, blue] (0, 0) — (4, 0); \draw[thick, blue] (4, 0) — (4, 3); \draw[thick, blue] (4, 3) — (0, 3); \draw[thick, blue] (0, 3) — (0, 0); \filldraw[blue] (0, 0) circle (2pt) node[below right] {$D(0, 0)$}; \filldraw[blue] (4, 0) circle (2pt) node[below right] {$E(c, 0)$}; \filldraw[blue] (4, 3) circle (2pt) node[above right] {$G(a, b)$}; \filldraw[blue] (0, 3) circle (2pt) node[above right] {$F(?, ?)$}; \end{tikzpicture} \end{array} \]

Solution

To find the coordinates of point \( F \) in parallelogram \( DEFG \), we need to use the properties of a parallelogram. Opposite sides are parallel and equal in length. From the diagram: - \( D = (0, 0) \) - \( E = (c, 0) \) - \( G = (a, b) \) - \( F = (?, ?) \) Since \( DEFG \) is a parallelogram, \( DE \parallel FG \) and \( DF \parallel EG \). Since \( DE \parallel FG \) and have the same length: \[ F_x - a = c - 0 \] Thus, \[ F_x = a + c \] Since \( DF \parallel EG \): \[ F_y - 0 = b - 0 \] Thus, \[ F_y = b \] Therefore, the coordinates of \( F \) are \((a + c, b)\).