Final answer:
The coordinates of vertex C in an isosceles trapezoid ABCD, with given vertices A(0,0), B(2,5), and D(8,0), are C(6,5). This is derived from the properties of an isosceles trapezoid and the reflection of point B across the line x=4, which is equidistant from A and D.
Step-by-step explanation:
The coordinates of vertex C in an isosceles trapezoid can be found by using the properties of this specific geometric shape. Since ABCD is an isosceles trapezoid, we know that line segments AB and CD will be parallel, and AD and BC will be of equal length due to the definition of an isosceles trapezoid where the non-parallel sides are equal in length.
Given vertices A(0,0), B(2,5), and D(8,0), and knowing that AD and BC must be equal, we can use the distance formula to find the length AD:
AD = √[(8-0)² + (0-0)²] = √[64] = 8 units.
Since BC must be equal to AD and angle A and angle D are right angles, we infer that triangle ABD is congruent to triangle DCA by the SAS postulate. Therefore, point C must be a vertical reflection of point B across the vertical line that would bisect segment AD, which is the line x = 4 (as it is the midpoint between A and D on the x-axis). So, the x-coordinate for point C will be 8 - 2 = 6 since point B is 2 units to the right of the line of symmetry (x=4), point C will be 2 units to the left of it.
The y-coordinate for point C will be the same as for point B since they are reflections across the vertical line x = 4. Therefore, the coordinates of point C will be C(6,5).