https://file.aiquickdraw.com/mathbot/file/172619094713220hlqsoa.jpg

To find the exact perimeter of pentagon \( ABCDE \), I will use the distance formula to calculate the lengths of each side of the pentagon. The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

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

### Coordinates of the vertices:
- \( A(0, 7) \)
- \( B(6, 0) \)
- \( C(3, 4) \)
- \( D(-3, -4) \)
- \( E(-6, 0) \)

We will now calculate the distances for each side:

1. \( AB \)
\[
AB = \sqrt{(6 - 0)^2 + (0 - 7)^2} = \sqrt{6^2 + (-7)^2} = \sqrt{36 + 49} = \sqrt{85}
\]

2. \( BC \)
\[
BC = \sqrt{(3 - 6)^2 + (4 - 0)^2} = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

3. \( CD \)
\[
CD = \sqrt{(3 - (-3))^2 + (4 - (-4))^2} = \sqrt{(6)^2 + (8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]

4. \( DE \)
\[
DE = \sqrt{(-6 - (-3))^2 + (0 - (-4))^2} = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

5. \( EA \)
\[
EA = \sqrt{(0 - (-6))^2 + (7 - 0)^2} = \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85}
\]

### Perimeter
The perimeter is the sum of all these distances:
\[
P = AB + BC + CD + DE + EA = \sqrt{85} + 5 + 10 + 5 + \sqrt{85}
\]
Simplifying:
\[
P = 2\sqrt{85} + 20
\]

Thus, the exact perimeter of pentagon \( ABCDE \) is:
\[
P = 2\sqrt{85} + 20
\]

Do you need more details or clarifications? Here are some related questions:

1. How is the distance formula derived?
2. Can the perimeter be simplified further?
3. What are some other methods to calculate the perimeter of polygons?
4. How would you approach finding the area of this pentagon?
5. Can you calculate the lengths of the diagonals of the pentagon?

**Tip:** Always verify the exact coordinates of vertices carefully to avoid calculation mistakes when using the distance formula!

Find the exact perimeter of pentagon ABCDE plotted below.

Learn how to calculate the exact perimeter of a pentagon using the distance formula. This problem involves coordinate geometry, calculating the distance between points, and finding the perimeter. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation