Identify the area of the polygon with vertices ,
, and

Ask by Black Mcfarlane. in the United States

Mar 23,2025

Question

Identify the area of the polygon with vertices $$ C(3,3) $$, $$ A(8,-2), S(3,-4) $$, and $$ H(0,-2) $$

UpStudy AI · Accepted Answer

**Step 1: List the vertices in order**

Let the vertices of the polygon be given as:
$$
C(3,3), \quad A(8,-2), \quad S(3,-4), \quad H(0,-2)
$$
We will assume the vertices are connected in the order $$ C \to A \to S \to H $$ and then back to $$ C $$.

**Step 2: Apply the Shoelace Formula**

The shoelace formula for the area $$ A $$ of a polygon with vertices $$(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$$ is given by:
$$
A = \frac{1}{2} \left| \sum_{i=1}^{n} \left( x_i y_{i+1} - y_i x_{i+1} \right) \right|
$$
where we set $$ (x_{n+1}, y_{n+1}) = (x_1, y_1) $$.

**Step 3: Substitute the coordinates**

Let:
$$
(x_1,y_1) = (3,3),\quad (x_2,y_2) = (8,-2),\quad (x_3,y_3) = (3,-4),\quad (x_4,y_4) = (0,-2)
$$
and $$ (x_5, y_5) = (3,3) $$.

Now, compute the required sums:

- Compute the sum $$ S_1 = x_1y_2 + x_2y_3 + x_3y_4 + x_4y_5 $$:
  $$
  S_1 = 3 \cdot (-2) + 8 \cdot (-4) + 3 \cdot (-2) + 0 \cdot 3 = -6 - 32 - 6 + 0 = -44
  $$

- Compute the sum $$ S_2 = y_1x_2 + y_2x_3 + y_3x_4 + y_4x_5 $$:
  $$
  S_2 = 3 \cdot 8 + (-2) \cdot 3 + (-4) \cdot 0 + (-2) \cdot 3 = 24 - 6 + 0 - 6 = 12
  $$

**Step 4: Calculate the area**

Now plug these into the formula:
$$
A = \frac{1}{2} \left| S_1 - S_2 \right| = \frac{1}{2} \left| -44 - 12 \right| = \frac{1}{2} \left| -56 \right| = \frac{56}{2} = 28
$$

**Final Answer:**

The area of the polygon is $$28$$.
The area of the polygon is 28.

Identify the area of the polygon with vertices ,
, and

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Identify the area of the polygon with vertices , , and