Question

learning target: i can find the fourth vertex of a square on the coordinate plane given two sides of the square. 17. determine point d such that quadrilateral abcd is a square. 18. determine point d such that quadrilateral abcd is a square.

Explanation:

Step1: Recall properties of a square

In a square, the diagonals bisect each other. Let the coordinates of \(A=(2,4)\), \(B=(8,0)\) and \(C=(4, - 6)\). The mid - point of the diagonal \(AC\) is the same as the mid - point of the diagonal \(BD\).
The mid - point formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
The mid - point of \(AC\) where \(x_1 = 2,y_1=4,x_2 = 4,y_2=-6\) is \(M_{AC}=(\frac{2 + 4}{2},\frac{4+( - 6)}{2})=(3,-1)\).
Let the coordinates of \(D=(x,y)\). The mid - point of \(BD\) with \(B=(8,0)\) is \(M_{BD}=(\frac{x + 8}{2},\frac{y + 0}{2})\).

Step2: Equate mid - points

Since \(M_{AC}=M_{BD}\), we have the following two equations:
\(\frac{x + 8}{2}=3\) and \(\frac{y+0}{2}=-1\).
For \(\frac{x + 8}{2}=3\), multiply both sides by 2: \(x + 8=6\), then \(x=6 - 8=-2\).
For \(\frac{y + 0}{2}=-1\), multiply both sides by 2: \(y=-2\).
So the coordinates of \(D\) are \((-2,-2)\).

Answer:

The coordinates of point \(D\) are \((-2,-2)\)