Question

the vertices of quadrilateral pqrs are listed.
\\p(3,7), q(6,-2), r(0,-4), s(-3,5)\\
which of the following is the strongest classification that identifies quadrilateral pqrs?

• quadrilateral pqrs is a parallelogram.
• quadrilateral pqrs is a square.
• quadrilateral pqrs is a rectangle.
• quadrilateral pqrs is a trapezoid.

Explanation:

⚡ Using what you learned: properties of quadrilaterals · slope of a line

Step 1: Find the slopes of the sides

Calculate the slopes of the four sides of quadrilateral \( PQRS \) using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):

• Slope of \( PQ \) (from \( P(3,7) \) to \( Q(6,-2) \)):

$$ m_{PQ} = \frac{-2 - 7}{6 - 3} = \frac{-9}{3} = -3 $$

• Slope of \( QR \) (from \( Q(6,-2) \) to \( R(0,-4) \)):

$$ m_{QR} = \frac{-4 - (-2)}{0 - 6} = \frac{-2}{-6} = \frac{1}{3} $$

• Slope of \( RS \) (from \( R(0,-4) \) to \( S(-3,5) \)):

$$ m_{RS} = \frac{5 - (-4)}{-3 - 0} = \frac{9}{-3} = -3 $$

• Slope of \( SP \) (from \( S(-3,5) \) to \( P(3,7) \)):

$$ m_{SP} = \frac{7 - 5}{3 - (-3)} = \frac{2}{6} = \frac{1}{3} $$

Step 2: Analyze parallel and perpendicular sides

• Parallel sides:

Since \( m_{PQ} = m_{RS} = -3 \), the opposite sides \( PQ \) and \( RS \) are parallel.
Since \( m_{QR} = m_{SP} = \frac{1}{3} \), the opposite sides \( QR \) and \( SP \) are parallel.
This confirms \( PQRS \) is at least a parallelogram.

• Perpendicular sides:

Compare the slopes of adjacent sides, such as \( PQ \) and \( QR \):

$$ m_{PQ} \cdot m_{QR} = -3 \cdot \frac{1}{3} = -1 $$

Since the product of their slopes is \(-1\), the adjacent sides are perpendicular, meaning all four angles are \( 90^\circ \). This makes \( PQRS \) at least a rectangle.

Step 3: Check side lengths to distinguish between rectangle and square

Calculate the lengths of adjacent sides \( PQ \) and \( QR \) using the distance formula:

• Length of \( PQ \):

$$ PQ = \sqrt{(6 - 3)^2 + (-2 - 7)^2} = \sqrt{3^2 + (-9)^2} = \sqrt{9 + 81} = \sqrt{90} $$

• Length of \( QR \):

$$ QR = \sqrt{(0 - 6)^2 + (-4 - (-2))^2} = \sqrt{(-6)^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} $$

Since the adjacent sides are not equal in length (\( \sqrt{90}
eq \sqrt{40} \)), the figure is not a square. The strongest classification is a rectangle.

Answer:

Quadrilateral PQRS is a rectangle.