Use the diagram to find the perimeter and the area of quadrilateral  $ABCD$ . Round your answers to the nearest hundredth.

Coordinate plane shows a polygon. The vertical axis extends from negative 6 to 5. Horizontal axis extends from negative 4 to 6. Coordinates of triangle A B F. Ordered pair A negative 5 comma 4. Ordered pair B 0 comma 3. Ordered pair F negative 2 comma 1. Coordinates of rectangle B C E F. Ordered pair B 0 comma 3. Ordered pair C 4 comma negative 1. Ordered pair E 2 comma negative 3. Ordered pair F negative 2 comma 1. Coordinates of triangle C D E. Ordered pair C 4 comma negative 1. Ordered pair D 4 comma negative 5. Ordered pair E 2 comma negative 3.

The perimeter is about  units.

The area is  square units.

To solve for the perimeter and the area of quadrilateral $ABCD$, we can use the coordinates provided and apply mathematical formulas.

### 1. Coordinates of the vertices:
The coordinates of the quadrilateral $ABCD$ are:
- $ A(-5, 4) $
- $ B(0, 3) $
- $ C(4, -1) $
- $ D(4, -5) $

### 2. Perimeter of the quadrilateral
The perimeter is the sum of the lengths of the sides of the quadrilateral. We will use the distance formula to calculate the lengths of the sides.

The distance formula between two points $ (x_1, y_1) $ and $ (x_2, y_2) $ is:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

#### Side $ AB $:
\[
AB = \sqrt{(0 - (-5))^2 + (3 - 4)^2} = \sqrt{(5)^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26} \approx 5.10
\]

#### Side $ BC $:
\[
BC = \sqrt{(4 - 0)^2 + (-1 - 3)^2} = \sqrt{(4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.66
\]

#### Side $ CD $:
\[
CD = \sqrt{(4 - 4)^2 + (-5 - (-1))^2} = \sqrt{(0)^2 + (-4)^2} = \sqrt{16} = 4
\]

#### Side $ DA $:
\[
DA = \sqrt{(4 - (-5))^2 + (-5 - 4)^2} = \sqrt{(9)^2 + (-9)^2} = \sqrt{81 + 81} = \sqrt{162} \approx 12.73
\]

Now, we can sum the lengths of the sides to find the perimeter:

\[
\text{Perimeter} = AB + BC + CD + DA = 5.10 + 5.66 + 4 + 12.73 = 27.49 \text{ units}
\]

### 3. Area of the quadrilateral
The area of a polygon with vertices $ (x_1, y_1), (x_2, y_2), \dots, (x_n, y_n) $ can be calculated using the **shoelace formula**:

\[
\text{Area} = \frac{1}{2} \left| x_1 y_2 + x_2 y_3 + \cdots + x_n y_1 - (y_1 x_2 + y_2 x_3 + \cdots + y_n x_1) \right|
\]

Using the coordinates of $ A(-5, 4), B(0, 3), C(4, -1), D(4, -5) $:

\[
\text{Area} = \frac{1}{2} \left| (-5)(3) + (0)(-1) + (4)(-5) + (4)(4) - \left(4(0) + 3(4) + (-1)(4) + (-5)(-5)\right) \right|
\]
\[
= \frac{1}{2} \left| -15 + 0 - 20 + 16 - \left(0 + 12 - 4 + 25\right) \right|
\]
\[
= \frac{1}{2} \left| -19 - 33 \right| = \frac{1}{2} \times 52 = 26 \text{ square units}
\]

### Final Answers:
- The perimeter of quadrilateral $ABCD$ is approximately **27.49 units**.
- The area of quadrilateral $ABCD$ is **26 square units**.

Would you like more details or have any questions? Here are some related questions:

1. How do we generalize the distance formula in 3D coordinates?
2. What are other methods to calculate the area of irregular polygons?
3. How do we derive the shoelace formula?
4. Can we use vector cross products to find the area of polygons?
5. What happens to the perimeter if we change one vertex position?

**Tip**: Always double-check your calculations by revisiting each step, especially in complex geometrical problems.

Use the diagram to find the perimeter and the area of quadrilateral ABCD. Round your answers to the nearest hundredth.

Learn how to find the perimeter and area of quadrilateral ABCD using the distance formula and shoelace theorem. Step-by-step solution provided for students in Grades 9-12, involving coordinates, geometry, and key formulas.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation