Quadrilateral PQRS is located at P (0, 1), Q (3, 2), R (4, 0), and S (1, −1). Russell and Jamie have both classified PQRS differently. Examine their proofs. Who is correct?
RUSSELL'S ANSWER
PQRS is a parallelogram because opposite sides are both congruent and parallel.
Segment PQ
P (0, 1) and Q (3, 2)
d equals the square root of the quantity 3 minus 0 all squared plus 2 minus 1 all squared equals the square root of the quantity 9 plus 1 equals the square root of 10
m equals 2 minus 1 over 3 minus 0 equals one third
Segment SR
S (1, −1) and R (4, 0)
d equals the square root of the quantity 4 minus 1 all squared plus 0 plus 1 all squared equals the square root of the quantity 9 plus 1 equals the square root of 10
m equals 0 plus 1 over 4 minus 1 equals one third
Segment PS
P (0, 1) and S (1, −1)
d equals the square root of the quantity 1 minus 0 all squared plus negative 1 minus 1 all squared equals the square root of the quantity 1 plus 4 equals the square root of 5
m equals negative 1 minus 1 over 1 minus 0 equals negative 2 over 1
Segment QR
Q (3, 2) and R (4, 0)
d equals the square root of the quantity 4 minus 3 all squared plus 0 minus 2 all squared equals the square root of the quantity 1 plus 4 equals the square root of 5
m equals 0 minus 2 over 4 minus 3 equals negative 2 over 1
JAIME'S ANSWER
Segments PQ and SR are both congruent and parallel, and segments PS and QR are both congruent and parallel. PQRS is a rectangle because opposite sides are both congruent and parallel and adjacent sides are perpendicular.
Segment PQ
P (0, 1) and Q (3, 2)
d equals the square root of the quantity 3 minus 0 all squared plus 2 minus 1 all squared equals the square root of the quantity 9 plus 1 equals the square root of 10
m equals 2 minus 1 over 3 minus 0 equals one third
Segment SR
S (1, −1) and R (4, 0)
d equals the square root of the quantity 4 minus 1 all squared plus 0 plus 1 all squared equals the square root of the quantity 9 plus 1 equals the square root of 10
m equals 0 plus 1 over 4 minus 1 equals one third
Segment PS
P (0, 1) and S (1, −1)
d equals the square root of the quantity 1 minus 0 all squared plus negative 1 minus 1 all squared equals the square root of the quantity 1 plus 4 equals the square root of 5
m equals negative 1 minus 1 over 1 minus 0 equals negative 2 over 1
Segment QR
Q (3, 2) and R (4, 0)
d equals the square root of the quantity 4 minus 3 all squared plus 0 minus 2 all squared equals the square root of the quantity 1 plus 4 equals the square root of 5
m equals 0 minus 2 over 4 minus 3 equals negative 2 over 1
Segments PQ and SR are both congruent and parallel, and segments PS and QR are both congruent and parallel. Segments PS and SR are perpendicular. Segments PQ and QR are perpendicular.
Russell
Jamie
Both
Neither
