Question

3 three vertices of quadrilateral abcd are given as shown in the diagram. what are the coordinates of vertex d if quadrilateral abcd is a square? (7,4) (6,5) (8,4) (7,6) clear all

Explanation:

Step1: Analyze the properties of a square

In a square, all sides are equal and adjacent sides are perpendicular.

Step2: Determine the side - length and direction

Assume \(A=(1,5)\), \(B=(0,0)\), \(C=(5, - 1)\). Calculate the vector from \(B\) to \(A\) as \((1 - 0,5 - 0)=(1,5)\). The vector from \(B\) to \(C\) is \((5 - 0,-1 - 0)=(5,-1)\).
For a square, if we consider the translation and rotation. If we assume the side - length and the right - angle property.
Counting the units from the graph (assuming each grid is 1 unit). The horizontal distance from \(A\) to \(B\) and the pattern of movement.
If we observe the horizontal and vertical displacements. From \(A\) to \(B\) and then from \(C\) to \(D\) should follow the same pattern of side - length and perpendicularity.
By visual inspection (since in a square, the distance between consecutive vertices is the same and angles are \(90^{\circ}\)).
If we consider the movement from \(C\) to \(D\) similar to the movement from \(A\) to \(B\) in terms of side - length and direction (adjusted for the square structure).
Counting the grids: The \(x\) - coordinate of \(D\) is \(5+(2)=7\) (since from \(B\) to \(A\) has a certain \(x\) - displacement and we replicate it for \(C\) to \(D\) considering the square's symmetry) and the \(y\) - coordinate of \(D\) is \(-1+(5)=4\)

Answer:

\((7,4)\)