3. Find the coordinates of a vertex \( D \) of the equadriateral \( A(0,4) B(-4,7) \) \( (-7,3) \) and \( \Delta \) if \( A D=B C \) and \( D \) is on \( x \)-akis.

To find the coordinates of vertex \( D \) of the quadrilateral with given points \( A(0,4) \), \( B(-4,7) \), and \( C(-7,3) \), we start by noting that \( D \) is on the x-axis, which means it has coordinates \( D(x, 0) \). Given the condition \( AD = BC \), we first calculate the lengths of the segments \( AD \) and \( BC \). The distance \( AD \) is determined by the formula: \[ AD = \sqrt{(x - 0)^2 + (0 - 4)^2} = \sqrt{x^2 + 16} \] Similarly, the distance \( BC \) is: \[ BC = \sqrt{(-7 - (-4))^2 + (3 - 7)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = 5 \] Setting these equal gives: \[ \sqrt{x^2 + 16} = 5 \] Squaring both sides results in: \[ x^2 + 16 = 25 \] \[ x^2 = 9 \] \[ x = 3 \text{ or } x = -3 \] Thus, the coordinates of vertex \( D \) can be either \( D(3, 0) \) or \( D(-3, 0) \). Therefore, the possible coordinates of vertex \( D \) of the equilateral quadrilateral could be either \( (3, 0) \) or \( (-3, 0) \).