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The polygon is a parallelogram. Its perimeter is \(4\sqrt{5} + 4\) and its area is 8.
Solución
We are given the vertices
\[
P(2,-2),\quad Q(4,2),\quad R(6,2),\quad S(4,-2)
\]
and we wish to identify the polygon, then compute its perimeter and its area.
---
**Step 1. Identify the Polygon**
Connect the vertices in order: \( P \to Q \to R \to S \to P \).
1. Compute vectors corresponding to the sides:
- \( \overrightarrow{PQ} = (4-2,\; 2-(-2)) = (2,4) \)
- \( \overrightarrow{QR} = (6-4,\; 2-2) = (2,0) \)
- \( \overrightarrow{RS} = (4-6,\; -2-2) = (-2,-4) \)
- \( \overrightarrow{SP} = (2-4,\; -2-(-2)) = (-2,0) \)
2. Notice that:
- \(\overrightarrow{PQ} = (2,4)\) and \(\overrightarrow{RS} = (-2,-4)\) have the same magnitude and are parallel (opposite in direction),
- \(\overrightarrow{QR} = (2,0)\) and \(\overrightarrow{SP} = (-2,0)\) are horizontal, equal in magnitude, and parallel (again oppositely directed).
3. In a quadrilateral, if both pairs of opposite sides are equal and parallel, the shape is a **parallelogram**.
Thus, the polygon is a parallelogram.
---
**Step 2. Find the Perimeter**
1. Find the lengths of each side:
- Length of \(PQ\):
\[
PQ = \sqrt{(2)^2 + (4)^2} = \sqrt{4+16} = \sqrt{20} = 2\sqrt{5}.
\]
- Length of \(QR\):
\[
QR = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2.
\]
- Length of \(RS\):
\[
RS = \sqrt{(-2)^2 + (-4)^2} = \sqrt{4+16} = \sqrt{20} = 2\sqrt{5}.
\]
- Length of \(SP\):
\[
SP = \sqrt{(-2)^2 + (0)^2} = \sqrt{4} = 2.
\]
2. The perimeter \(P\) is the sum of the side lengths:
\[
P = PQ+QR+RS+SP = 2\sqrt{5} + 2 + 2\sqrt{5} + 2 = 4\sqrt{5} + 4.
\]
---
**Step 3. Find the Area**
One way to find the area of a parallelogram is by using the magnitude of the cross product of two adjacent side vectors. Choose vertex \(P\) with adjacent sides \(\overrightarrow{PQ}\) and \(\overrightarrow{PS}\).
1. We have:
- \(\overrightarrow{PQ} = (2,4)\),
- \(\overrightarrow{PS} = S - P = (4-2,\; -2-(-2)) = (2,0)\).
2. The area \(A\) is given by the absolute value of the determinant:
\[
A = \left| \det\begin{pmatrix} 2 & 4 \\ 2 & 0 \end{pmatrix} \right| = \left| (2)(0) - (4)(2) \right| = |0-8| = 8.
\]
---
**Final Answers:**
- The polygon is a **parallelogram**.
- The perimeter is
\[
4\sqrt{5}+4.
\]
- The area is
\[
8.
\]
Respondido por UpStudy AI y revisado por un tutor profesional

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