The area of the polygon PQRS is 25 square units ⇒ answer D
Step-by-step explanation:
Let us revise some important notes
The vertices of polygon PQRS are:
P = (1 , -1)
Q = (6 , -1)
R = (6 , -6)
S = (1 , -6)
∵ The polygon has 4 vertices
∴ The polygon PQRS is a quadrilateral
∵ y-coordinates of points P and Q are equal
∴ PQ is a horizontal side
∵ The length of PQ = [tex]x_{Q}-x_{P}[/tex]
∵ [tex]x_{Q}=6[/tex] and [tex]x_{P}=1[/tex]
∴ PQ = 6 - 1 = 5 units
∵ x-coordinates of points Q and R are equal
∴ QR is a vertical side
∵ The length of QR = [tex]y_{Q}-y_{R}[/tex]
∵ [tex]y_{Q}=-1[/tex] and [tex]y_{R}=-6[/tex]
∴ QR = -1 - (-6) = -1 + 6 = 5 units
∵ y-coordinates of points R and S are equal
∴ RS is a horizontal side
∵ The length of RS = [tex]x_{R}-x_{S}[/tex]
∵ [tex]x_{R}=6[/tex] and [tex]x_{S}=1[/tex]
∴ RS = 6 - 1 = 5 units
∵ x-coordinates of points S and P are equal
∴ SP is a vertical side
∵ The length of SP = [tex]y_{P}-y_{S}[/tex]
∵ [tex]y_{P}=-1[/tex] and [tex]y_{S}=-6[/tex]
∴ SP = -1 - (-6) = -1 + 6 = 5 units
In the quadrilateral PQRS
∵ PQ and SR are horizontal lines
∵ PQ // SR
∵ QR and SP are vertical lines
∴ QR // SP
∵ Each horizontal line ⊥ to each vertical line
∴ Each two adjacent sides are ⊥
∴ PQ ⊥ QR , QR ⊥ RS , RS ⊥ SP , SP ⊥ PQ
∴ m∠P = 90° , m∠Q = 90° , m∠R = 90° , m∠S = 90°
∵ PQ = QR RS = SP = 5 units
∵ Each two opposite sides are parallel
∵ All sides are equal in length
∵ All angles are right angles
∴ The quadrilateral PQRS is a square of side length 5 units
∵ Area of square = (side)²
∵ The length of the side is 5 units
∴ Area of square PQRS = (5)² = 25 units²
The area of the polygon PQRS is 25 square units
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