Identify the area of the polygon with vertices ,
, and .

Ask by Wang Vaughan. in the United States

Mar 21,2025

Question

Identify the area of the polygon with vertices $$ P(1,2) $$, $$ Q(1,4), R(-1,6) $$, and $$ S(-3,2) $$.

UpStudy AI · Accepted Answer

To find the area of the polygon with vertices

,

, and

, we can use the shoelace formula:

Plugging in the coordinates:

Calculating the terms for the formula:

Calculating the sum:

Sum of products for the first part:

Now for the second part:

Sum of products for the second part:

Putting it all together:

Thus, the area of the polygon is

.

The Shoelace formula is given by:

$$ \text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - x_{i+1} y_i) \right| $$

where $$ (x_i, y_i) $$ are the coordinates of the vertices of the polygon, and $$ n $$ is the number of vertices.

Let's calculate the area of the polygon using the Shoelace formula.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1}{2}\left|\left(1\times 4-1\times 2\right)+\left(1\times 6-\left(-1\right)\times 4\right)+\left(\left(-1\right)\times 2-\left(-3\right)\times 6\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|$$
- step1: Calculate:
  $$\frac{1}{2}\left|\left(4-1\times 2\right)+\left(1\times 6-\left(-1\right)\times 4\right)+\left(\left(-1\right)\times 2-\left(-3\right)\times 6\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|$$
- step2: Multiply the numbers:
  $$\frac{1}{2}\left|\left(4-2\right)+\left(1\times 6-\left(-1\right)\times 4\right)+\left(\left(-1\right)\times 2-\left(-3\right)\times 6\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|$$
- step3: Subtract the numbers:
  $$\frac{1}{2}\left|2+\left(1\times 6-\left(-1\right)\times 4\right)+\left(\left(-1\right)\times 2-\left(-3\right)\times 6\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|$$
- step4: Remove the parentheses:
  $$\frac{1}{2}\left|2+\left(1\times 6-\left(-4\right)\right)+\left(\left(-1\right)\times 2-\left(-3\right)\times 6\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|$$
- step5: Calculate:
  $$\frac{1}{2}\left|2+\left(6-\left(-4\right)\right)+\left(\left(-1\right)\times 2-\left(-3\right)\times 6\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|$$
- step6: Subtract the terms:
  $$\frac{1}{2}\left|2+10+\left(\left(-1\right)\times 2-\left(-3\right)\times 6\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|$$
- step7: Remove the parentheses:
  $$\frac{1}{2}\left|2+10+\left(-2-\left(-3\times 6\right)\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|$$
- step8: Multiply the numbers:
  $$\frac{1}{2}\left|2+10+\left(-2-\left(-18\right)\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|$$
- step9: Subtract the terms:
  $$\frac{1}{2}\left|2+10+16+\left(\left(-3\right)\times 2-1\times 6\right)\right|$$
- step10: Remove the parentheses:
  $$\frac{1}{2}\left|2+10+16+\left(-3\times 2-1\times 6\right)\right|$$
- step11: Multiply the numbers:
  $$\frac{1}{2}\left|2+10+16+\left(-6-1\times 6\right)\right|$$
- step12: Multiply the numbers:
  $$\frac{1}{2}\left|2+10+16+\left(-6-6\right)\right|$$
- step13: Subtract the numbers:
  $$\frac{1}{2}\left|2+10+16+\left(-12\right)\right|$$
- step14: Remove the parentheses:
  $$\frac{1}{2}\left|2+10+16-12\right|$$
- step15: Calculate:
  $$\frac{1}{2}\left|16\right|$$
- step16: Calculate the absolute value:
  $$\frac{1}{2}\times 16$$
- step17: Reduce the numbers:
  $$1\times 8$$
- step18: Simplify:
  $$8$$
The area of the polygon is 10 square units.

Identify the area of the polygon with vertices ,
, and .

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Identify the area of the polygon with vertices , , and .