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Answer:
PQ = RS = 5√2 units
QR = SP = 4√2 units
Step-by-step explanation:
It is given that The coordinates of the vertices are P(2, -2) , Q(7, 3) , R(11, -1) ,
and S(6, -6)
We have to find the length of PQ, QR, RS, and SP.
For PQRS to be a parallelogram,
PQ = RS and QR = SP
The distance between two points is calculated by using distance formula,
√(x₂ - x₁)² + (y₂ - y₁)²
PQ = √(x₂ - x₁)² + (y₂ - y₁)²
P(2, -2) and Q(7, 3)
Here, x₁ = 2, x₂ = 7, y₁ = -2, and y₂ = 3
PQ = √(x₂ - x₁)² + (y₂ - y₁)²
= √(7 - 2)² + (3 - (-2))² = √(5)² + (5)²
=√25 + 25 = √50 = 5√2 units
PQ = 5√2 units
QR = √(x₂ - x₁)² + (y₂ - y₁)²
Q(7, 3) and R(11, -1)
Here, x₁ = 7, x₂ = 11, y₁ = 3, and y₂ = -1
QR = √(x₂ - x₁)² + (y₂ - y₁)²
= √(11 - 7)² + ((-1) - 3)² = √(4)² + (-4)²
=√16 + 16 = √32 = 4√2 units
QR = 4√2 units
RS = √(x₂ - x₁)² + (y₂ - y₁)²
R(11, -1) and S(6, -6)
Here, x₁ = 11, x₂ = 6, y₁ = -1, and y₂ = -6
RS = √(x₂ - x₁)² + (y₂ - y₁)²
= √(6 - 11)² + ((-6) - (-1))² = √(-5)² + (-5)²
=√25 + 25 = √50 = 5√2 units
RS = 5√2 units
SP = √(x₂ - x₁)² + (y₂ - y₁)²
S(6, -6) and R(2, -2)
Here, x₁ = 6, x₂ = 2, y₁ = -6, and y₂ = -2
SP = √(x₂ - x₁)² + (y₂ - y₁)²
= √(2 - 6)² + ((-2) - (-6))² = √(-4)² + (4)²
=√16 + 16 = √32 = 4√2 units
SP = 4√2 units
Now, we know that PQ = RS = 5√2 units
and QR = SP = 4√2 units
So, PQRS is a parallelogram.