Question

part 3: calculating perimeter of polygons and classifying
find the perimeter of each polygon. if it’s a quadrilatera
ectangle, or trapezoid.
13.
perimeter:
slope of pq =
slope of sr =
slope of ps =
slope of qr =
relationship between the slopes is

Explanation:

Step1: Identify Coordinates

Assume each grid square has side length 1. Let's find coordinates:

• \( P(-6, 2) \), \( Q(-2, 6) \), \( R(2, 2) \), \( S(-2, -2) \)

Step2: Calculate Side Lengths (Distance Formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \))

• \( PQ \): \( \sqrt{(-2 + 6)^2 + (6 - 2)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \)
• \( QR \): \( \sqrt{(2 + 2)^2 + (2 - 6)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \)
• \( RS \): \( \sqrt{(-2 - 2)^2 + (-2 - 2)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \)
• \( SP \): \( \sqrt{(-6 + 2)^2 + (2 + 2)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \)

Step3: Calculate Perimeter

Perimeter = \( PQ + QR + RS + SP = 4\sqrt{2} + 4\sqrt{2} + 4\sqrt{2} + 4\sqrt{2} = 16\sqrt{2} \approx 22.63 \) (or exact form \( 16\sqrt{2} \))

Step4: Calculate Slopes (Slope Formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \))

• Slope of \( PQ \): \( \frac{6 - 2}{-2 + 6} = \frac{4}{4} = 1 \)
• Slope of \( SR \): \( \frac{2 - (-2)}{2 - (-2)} = \frac{4}{4} = 1 \) (Wait, correction: \( S(-2, -2) \), \( R(2, 2) \): \( \frac{2 + 2}{2 + 2} = 1 \))
• Slope of \( PS \): \( \frac{-2 - 2}{-2 + 6} = \frac{-4}{4} = -1 \)
• Slope of \( QR \): \( \frac{2 - 6}{2 + 2} = \frac{-4}{4} = -1 \)

Step5: Analyze Slope Relationships

• \( PQ \parallel SR \) (slopes both 1), \( PS \parallel QR \) (slopes both -1). Product of slopes of perpendicular sides: \( 1 \times (-1) = -1 \), so adjacent sides are perpendicular.

Step6: Classify Polygon

Since all sides equal and adjacent sides perpendicular, it's a square (a type of rectangle).

Answer:

Perimeter: \( 16\sqrt{2} \) (or ≈22.63)
Slope of \( PQ = 1 \), Slope of \( SR = 1 \), Slope of \( PS = -1 \), Slope of \( QR = -1 \)
Relationship: \( PQ \parallel SR \), \( PS \parallel QR \); \( PQ \perp PS \), \( PS \perp SR \), etc.
Polygon: Square (or Rectangle)