1. **Problem statement:** We have a square ABCF and a parallelogram CDEF on the Cartesian plane. Given points F(10,12), D(26,10), and E(18,0), we need to find the coordinates of point B. 2. **Key properties:** - ABCF is a square, so all sides are equal and adjacent sides are perpendicular. - CDEF is a parallelogram, so opposite sides are parallel and equal in length. - Points C and F are shared by both shapes. 3. **Find point C using parallelogram properties:** In parallelogram CDEF, vector \(\overrightarrow{CD} = \overrightarrow{EF}\). Given \(D(26,10)\) and \(E(18,0)\), vector \(\overrightarrow{EF} = F - E = (10 - 18, 12 - 0) = (-8, 12)\). So, \(\overrightarrow{CD} = (-8, 12)\). Since \(D = C + \overrightarrow{CD}\), then \(C = D - \overrightarrow{CD} = (26,10) - (-8,12) = (26 + 8, 10 - 12) = (34, -2)\). 4. **Find point B using square properties:** Square ABCF has points A, B, C, F in order. We know F(10,12) and C(34,-2). In a square, \(\overrightarrow{AB} = \overrightarrow{FC}\) and \(\overrightarrow{BC} = \overrightarrow{AF}\). We want B, so use \(\overrightarrow{BF} = \overrightarrow{AC}\) or equivalently, since ABCF is a square, the vector from F to C is perpendicular to vector from F to B and equal in length. Calculate vector \(\overrightarrow{FC} = C - F = (34 - 10, -2 - 12) = (24, -14)\). Since ABCF is a square, \(\overrightarrow{FB}\) is perpendicular to \(\overrightarrow{FC}\) and has the same length. Length of \(\overrightarrow{FC} = \sqrt{24^2 + (-14)^2} = \sqrt{576 + 196} = \sqrt{772} = 2\sqrt{193}\). A vector perpendicular to \((24, -14)\) is \((14, 24)\) or \((-14, -24)\). Check length of \((14, 24)\): \(\sqrt{14^2 + 24^2} = \sqrt{196 + 576} = \sqrt{772} = 2\sqrt{193}\), same length. So, \(\overrightarrow{FB} = (14, 24)\). Therefore, \(B = F + \overrightarrow{FB} = (10, 12) + (14, 24) = (24, 36)\). 5. **Final answer:** The coordinates of point B are \(\boxed{(24, 36)}\).