Question

parallelogram (abcd) has the following vertices:

• (a(0,8))
• (b(8,4))
• (c(2,-8))
• (d(-6,-4))

is parallelogram (abcd) a rectangle, and why?

choose 1 answer:

a yes, because (ab = ad) and (bc = cd), and (abcd) is a parallelogram.

b yes, because (overline{bc}) is perpendicular to (overline{ab}), and (abcd) is a parallelogram.

c no, because (overline{ab}) is shorter than (overline{ad}).

d no, because (overline{bc}) is not perpendicular to (overline{ab}).

Explanation:

Step1: Calculate the slope of \(AB\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(A(0,8)\) and \(B(8,4)\), \(m_{AB}=\frac{4 - 8}{8 - 0}=\frac{-4}{8}=-\frac{1}{2}\)

Step2: Calculate the slope of \(BC\)

For points \(B(8,4)\) and \(C(2,-8)\), \(m_{BC}=\frac{-8 - 4}{2 - 8}=\frac{-12}{-6} = 2\)

Step3: Check perpendicularity

Two lines with slopes \(m_1\) and \(m_2\) are perpendicular if \(m_1\times m_2=- 1\). Here, \(m_{AB}\times m_{BC}=-\frac{1}{2}\times2=-1\)

Answer:

B. Yes, because \(\overline{BC}\) is perpendicular to \(\overline{AB}\), and \(ABCD\) is a parallelogram.