Compute the perimeter and area of the quadrilateral determined by the points $$ A(1,-2), B(5,-2) $$, $$ C(5,3) $$, and $$ D(1,3) $$. $$ P=\square $$ unit $$ A=\square $$ unit $$ ^{2} $$ Question Help: Video Submit Question

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UpStudy AI · Accepted Answer

**Step 1. Calculate the lengths of the sides**

- For side $$ AB $$ (from $$ A(1,-2) $$ to $$ B(5,-2) $$):
  $$
  AB = \sqrt{(5-1)^2 + (-2 - (-2))^2} = \sqrt{4^2 + 0^2} = 4
  $$

- For side $$ BC $$ (from $$ B(5,-2) $$ to $$ C(5,3) $$):
  $$
  BC = \sqrt{(5-5)^2 + (3 - (-2))^2} = \sqrt{0^2 + 5^2} = 5
  $$

- For side $$ CD $$ (from $$ C(5,3) $$ to $$ D(1,3) $$):
  $$
  CD = \sqrt{(1-5)^2 + (3-3)^2} = \sqrt{(-4)^2 + 0^2} = 4
  $$

- For side $$ DA $$ (from $$ D(1,3) $$ to $$ A(1,-2) $$):
  $$
  DA = \sqrt{(1-1)^2 + (-2-3)^2} = \sqrt{0^2 + (-5)^2} = 5
  $$

**Step 2. Compute the Perimeter**

Sum the lengths of all sides:
$$
P = AB + BC + CD + DA = 4 + 5 + 4 + 5 = 18 \quad \text{units}
$$

**Step 3. Compute the Area**

Since the quadrilateral is a rectangle, the area is the product of its length and width. The horizontal distance (length) is:
$$
5 - 1 = 4 \quad \text{units}
$$
and the vertical distance (width) is:
$$
3 - (-2) = 5 \quad \text{units}
$$
Thus, the area is:
$$
A = 4 \times 5 = 20 \quad \text{square units}
$$

**Final Answer:**  
$$P = 18$$ unit and $$A = 20$$ unit$$^2$$.
The perimeter is 18 units and the area is 20 square units.

Compute the perimeter and area of the quadrilateral determined by the points \( A(1,-2), B(5,-2) \), \( C(5,3) \), and \( D(1,3) \). \( P=\square \) unit \( A=\square \) unit \( ^{2} \) Question Help: Video Submit Question

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