Question

rectangles have four right angles and opposite sides that are parallel.
a. is the figure shown a rectangle? explain.
b. if not, how could the points change so it would be a rectangle?

Explanation:

Step1: Find the coordinates of the points

Let \(A(-1,2)\), \(B(-1,-3)\), \(C(5,-4)\), \(D(6,1)\)

Step2: Calculate the slope of \(\overline{AB}\)

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For \(A(-1,2)\) and \(B(-1,-3)\), \(m_{AB}=\frac{-3 - 2}{-1-(-1)}=\text{undefined}\) (vertical line)

Step3: Calculate the slope of \(\overline{CD}\)

For \(C(5,-4)\) and \(D(6,1)\), \(m_{CD}=\frac{1-(-4)}{6 - 5}=5\)
Since \(m_{AB}
eq m_{CD}\), \(\overline{AB}\) and \(\overline{CD}\) are not parallel

Step4: Calculate the slope of \(\overline{BC}\)

For \(B(-1,-3)\) and \(C(5,-4)\), \(m_{BC}=\frac{-4-(-3)}{5-(-1)}=-\frac{1}{6}\)

Step5: Calculate the slope of \(\overline{DA}\)

For \(D(6,1)\) and \(A(-1,2)\), \(m_{DA}=\frac{2 - 1}{-1-6}=-\frac{1}{7}\)
Since \(m_{BC}
eq m_{DA}\), \(\overline{BC}\) and \(\overline{DA}\) are not parallel

Step6: Check for perpendicularity

For two lines to be perpendicular, \(m_1\times m_2=- 1\). Since slopes of adjacent sides do not satisfy this (e.g., \(m_{AB}\) (undefined) and \(m_{BC}=-\frac{1}{6}\) do not give product \(-1\)), there are no right - angles

Answer:

The slope of \(\overline{AB}\) is \(\text{undefined}\) and the slope of \(\overline{CD}\) is \(5\), so \(\overline{AB}\) and \(\overline{CD}\) are not parallel. The slope of \(\overline{BC}\) is \(-\frac{1}{6}\) and the slope of \(\overline{DA}\) is \(-\frac{1}{7}\), so \(\overline{BC}\) and \(\overline{DA}\) are not parallel. These slopes also show that no adjacent sides are perpendicular, and so there is no right angle. The figure shown is not a rectangle.