Question

2
select the correct answer from each drop-down menu.
the coordinates of a quadrilateral are (2,1), (-1,3), (-5,-3), and (-2,-5).
the quadrilateral is a rectangle because both pairs of opposite sides are parallel and since the product of the slopes of both pairs of segments is -
the adjacent sides are perpendicular
the angles formed by adjacent sides are greater than 90°
next

Explanation:

Step1: Recall Slope Formula

The slope \( m \) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). We calculate slopes of all sides.

• Side 1: Between \((2,1)\) and \((-1,3)\): \( m_1=\frac{3 - 1}{-1 - 2}=\frac{2}{-3}=-\frac{2}{3} \)
• Side 2: Between \((-1,3)\) and \((-5,-3)\): \( m_2=\frac{-3 - 3}{-5 - (-1)}=\frac{-6}{-4}=\frac{3}{2} \)
• Side 3: Between \((-5,-3)\) and \((-2,-5)\): \( m_3=\frac{-5 - (-3)}{-2 - (-5)}=\frac{-2}{3}=-\frac{2}{3} \)
• Side 4: Between \((-2,-5)\) and \((2,1)\): \( m_4=\frac{1 - (-5)}{2 - (-2)}=\frac{6}{4}=\frac{3}{2} \)

Step2: Check Parallel Sides

Slopes of side 1 (\(-\frac{2}{3}\)) and side 3 (\(-\frac{2}{3}\)) are equal, so they are parallel. Slopes of side 2 (\(\frac{3}{2}\)) and side 4 (\(\frac{3}{2}\)) are equal, so they are parallel.

Step3: Check Perpendicular Sides

The product of slopes of adjacent sides (e.g., \( m_1 \times m_2 = -\frac{2}{3} \times \frac{3}{2} = -1 \)) and (\( m_2 \times m_3 = \frac{3}{2} \times -\frac{2}{3} = -1 \)) and so on. When product of slopes is \(-1\), lines are perpendicular. So adjacent sides are perpendicular.

Step4: Analyze the Drop - down

The first part says the quadrilateral is a rectangle (since opposite sides parallel and adjacent perpendicular). The second drop - down is about why it's a rectangle. The option "the adjacent sides are perpendicular" is correct because for a rectangle, adjacent sides (which are sides of the rectangle, forming right angles) should be perpendicular (since product of their slopes is \(-1\)). The other option about angles greater than \(90^\circ\) is wrong as in a rectangle angles are \(90^\circ\) (right angles, so adjacent sides perpendicular).

Answer:

The correct option for the drop - down is "the adjacent sides are perpendicular"