Question

find the coordinate point for c that would make abcd a rhombus. (3, 5) (5, 1) (1, 1) (2, 4)

Explanation:

Step1: Recall rhombus property

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

Step2: Find mid - point of \(BD\)

For \(B=(2,3)\) and \(D=(4,3)\), \(x_1 = 2,y_1=3,x_2 = 4,y_2 = 3\).
The mid - point of \(BD\) is \((\frac{2 + 4}{2},\frac{3+3}{2})=(3,3)\).

Step3: Let the coordinates of \(C=(x,y)\) and find \(x\) and \(y\) using mid - point of \(AC\)

For \(A=(3,1)\) and \(C=(x,y)\), the mid - point is \((\frac{3 + x}{2},\frac{1 + y}{2})\).
Since the mid - point of \(AC\) is the same as the mid - point of \(BD\) (which is \((3,3)\)).
We have \(\frac{3 + x}{2}=3\) and \(\frac{1 + y}{2}=3\).
From \(\frac{3 + x}{2}=3\), we get \(3+x = 6\), so \(x=3\).
From \(\frac{1 + y}{2}=3\), we get \(1 + y=6\), so \(y = 5\).

Answer:

\((3,5)\)