Question

select the correct answer.
the vertices of a parallelogram are ( a(x_1,y_1) ), ( b(x_2,y_2) ), ( c(x_3,y_3) ), and ( d(x_4,y_4) ). which of the following must be true if parallelogram ( abcd ) is proven to be a rectangle?
a. ( (\frac{y_4 - y_1}{x_4 - x_3}=\frac{y_3 - y_2}{x_3 - x_2}) ) and ( (\frac{y_4 - y_1}{x_4 - x_1}\times\frac{y_3 - y_2}{x_3 - x_2})=-1 )
b. ( (\frac{y_4 - y_3}{x_4 - x_3}=\frac{y_2 - y_1}{x_2 - x_1}) ) and ( (\frac{y_4 - y_3}{x_4 - x_3}\times\frac{y_2 - y_1}{x_2 - x_1})=-1 )
c. ( (\frac{y_4 - y_3}{x_4 - x_3}=\frac{y_2 - y_1}{x_2 - x_1}) ) and ( (\frac{y_4 - y_1}{x_4 - x_3}\times\frac{y_3 - y_2}{x_3 - x_2})=-1 )
d. ( (\frac{y_4 - y_3}{x_4 - x_3}=\frac{y_3 - y_1}{x_3 - x_1}) ) and ( (\frac{y_4 - y_3}{x_4 - x_3}\times\frac{y_2 - y_1}{x_2 - x_1})=-1 )

Explanation:

Step1: Recall the property of a rectangle

In a rectangle, adjacent sides are perpendicular. The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). If two lines with slopes \(m_1\) and \(m_2\) are perpendicular, then \(m_1\times m_2=- 1\).

Step2: Analyze the slopes of adjacent sides of parallelogram \(ABCD\)

For parallelogram \(ABCD\), \(AB\parallel CD\) and \(AD\parallel BC\). The slope of \(AB\) is \(m_{AB}=\frac{y_2 - y_1}{x_2 - x_1}\), the slope of \(BC\) is \(m_{BC}=\frac{y_3 - y_2}{x_3 - x_2}\), the slope of \(CD\) is \(m_{CD}=\frac{y_4 - y_3}{x_4 - x_3}\), and the slope of \(DA\) is \(m_{DA}=\frac{y_1 - y_4}{x_1 - x_4}\). Since \(AB\parallel CD\) (\(m_{AB}=m_{CD}=\frac{y_2 - y_1}{x_2 - x_1}=\frac{y_4 - y_3}{x_4 - x_3}\)) and for \(AB\perp BC\), we need \(m_{AB}\times m_{BC}=-1\) (i.e., \(\frac{y_2 - y_1}{x_2 - x_1}\times\frac{y_3 - y_2}{x_3 - x_2}=-1\)) or \(m_{BC}\times m_{CD}=-1\) (equivalent to \(\frac{y_4 - y_3}{x_4 - x_3}\times\frac{y_3 - y_2}{x_3 - x_2}=-1\))

Answer:

B. \((\frac{y_4 - y_3}{x_4 - x_3}=\frac{y_2 - y_1}{x_2 - x_1})\) and \((\frac{y_4 - y_3}{x_4 - x_3}\times\frac{y_3 - y_2}{x_3 - x_2}=-1)\)