Question

quadrilateral pqrs is located on the coordinate plane. the lengths and slopes of the sides have been calculated
segment pq
d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
m = \frac{4 - 0}{3 - 0} = \frac{4}{3}
segment qr
d = \sqrt{(8 - 3)^2 + (4 - 4)^2} = \sqrt{25} = 5
m = \frac{4 - 4}{8 - 3} = 0
segment rs
d = \sqrt{(5 - 8)^2 + (0 - 4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
m = \frac{0 - 4}{5 - 8} = \frac{4}{3}
segment sp
d = \sqrt{(0 - 5)^2 + (0 - 0)^2} = \sqrt{25} = 5
m = \frac{0 - 0}{0 - 5} = 0
what is the best classification for quadrilateral pqrs?
rhombus
square
rectangle
parallelogram

Explanation:

Step1: Analyze side lengths

From the given data, \( d_{PQ} = 5 \), \( d_{QR} = 5 \), \( d_{RS} = 5 \), \( d_{SP} = 5 \). So all four sides are equal in length.

Step2: Analyze slopes (parallelism)

Slope of \( PQ \): \( m_{PQ}=\frac{4 - 0}{3 - 0}=\frac{4}{3} \)
Slope of \( QR \): \( m_{QR}=\frac{4 - 4}{8 - 3}=0 \)
Slope of \( RS \): \( m_{RS}=\frac{0 - 4}{5 - 8}=\frac{4}{3} \)
Slope of \( SP \): \( m_{SP}=\frac{0 - 0}{0 - 5}=0 \)

We can see that \( m_{PQ}=m_{RS}=\frac{4}{3} \) (so \( PQ \parallel RS \)) and \( m_{QR}=m_{SP} = 0 \) (so \( QR \parallel SP \)). So opposite sides are parallel.

Step3: Analyze slopes (perpendicularity)

For two lines with slopes \( m_1 \) and \( m_2 \), they are perpendicular if \( m_1\times m_2=- 1 \). Let's check \( m_{PQ}\times m_{QR}=\frac{4}{3}\times0 = 0
eq - 1 \), \( m_{QR}\times m_{RS}=0\times\frac{4}{3}=0
eq - 1 \), etc. So adjacent sides are not perpendicular.

A rhombus is a quadrilateral with all sides equal in length and opposite sides parallel (adjacent sides not necessarily perpendicular). A square has all sides equal and adjacent sides perpendicular, a rectangle has opposite sides equal and adjacent sides perpendicular, a parallelogram has opposite sides parallel (not necessarily all sides equal). Since all sides are equal and opposite sides are parallel (adjacent sides not perpendicular), the quadrilateral is a rhombus.

Answer:

Rhombus