Identify the polygon with vertices $$ P(2,-2), Q(4,2) $$, $$ R(6,2) $$, and $$ S(4,-2) $$, and then find the perimeter and area of the polygon. rhombus; $$ P=16 $$ units; $$ A=21 $$ units $$ ^{2} $$ parallelogram; $$ P=(4+4 \sqrt{5}) $$ units; $$ A=8 $$ units $$ ^{2} $$ rhombus; $$ P=21 $$ units; $$ A=16 $$ units $$ ^{2} $$ parallelogram; $$ P=8 $$ units; $$ A=(4+4 \sqrt{5}) $$ units 2

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UpStudy AI · Accepted Answer

**Step 1. Plotting and Connecting the Points**

We have the vertices  
$$
P(2,-2),\quad Q(4,2),\quad R(6,2),\quad S(4,-2)
$$  
Assuming the vertices are connected in the order $$ P \to Q \to R \to S \to P $$, the figure is a quadrilateral.

**Step 2. Identify the Quadrilateral**

Compute the vectors along the sides:
- $$\vec{PQ} = Q - P = (4-2,\,2-(-2)) = (2,\,4)$$
- $$\vec{QR} = R - Q = (6-4,\,2-2) = (2,\,0)$$
- $$\vec{RS} = S - R = (4-6,\,-2-2) = (-2,\,-4)$$
- $$\vec{SP} = P - S = (2-4,\,-2-(-2)) = (-2,\,0)$$

Notice that:
- $$\vec{PQ}$$ and $$\vec{RS}$$ are opposites: $$\vec{RS} = -\vec{PQ}$$
- $$\vec{QR}$$ and $$\vec{SP}$$ are opposites: $$\vec{SP} = -\vec{QR}$$

Because opposite sides are parallel and equal in length, the quadrilateral is a **parallelogram**.

**Step 3. Compute the Side Lengths**

- Length of $$PQ$$: 
  $$
  |\vec{PQ}| = \sqrt{2^2+4^2} = \sqrt{4+16} = \sqrt{20} = 2\sqrt{5}
  $$
- Length of $$QR$$:
  $$
  |\vec{QR}| = \sqrt{2^2+0^2} = 2
  $$

Since opposite sides are equal, the perimeter $$P$$ is given by:
$$
P = 2\left(2\sqrt{5} + 2\right) = 4\sqrt{5} + 4
$$

**Step 4. Compute the Area**

For a parallelogram, one way to compute the area is to find the magnitude of the cross product of two adjacent side vectors. Using $$\vec{PQ}$$ and $$\vec{QR}$$:
$$
\text{Area} = \left|\vec{PQ} \times \vec{QR}\right| = \left|\begin{vmatrix} 2 & 4 \ 2 & 0 \end{vmatrix}\right| = |2\cdot0 - 4\cdot2| = |-8| = 8
$$

**Conclusion**

The polygon is a **parallelogram** with:

- Perimeter: $$4 + 4\sqrt{5}$$ units,
- Area: $$8$$ square units.
The polygon is a parallelogram with a perimeter of $$4 + 4\sqrt{5}$$ units and an area of $$8$$ square units.

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