Find the perimeter and area of the polygon with the given vertices.

E(6, −2), F(6, 5), G(−1, 5)

To find the perimeter and area of the polygon formed by the vertices \( E(6, -2) \), \( F(6, 5) \), and \( G(-1, 5) \), we first recognize that this is a right triangle because two sides are parallel to the coordinate axes.

### Step 1: Find the side lengths
The polygon is a triangle with three sides, so we need to find the lengths of each side.

#### Length of \( EF \):
Points \( E(6, -2) \) and \( F(6, 5) \) have the same x-coordinate, which means the segment between them is vertical. The length is the difference in their y-coordinates:
\[
EF = |5 - (-2)| = |5 + 2| = 7
\]

#### Length of \( FG \):
Points \( F(6, 5) \) and \( G(-1, 5) \) have the same y-coordinate, which means the segment between them is horizontal. The length is the difference in their x-coordinates:
\[
FG = |6 - (-1)| = |6 + 1| = 7
\]

#### Length of \( EG \):
We use the distance formula to find the length of \( EG \), as this is the diagonal side of the triangle:
\[
EG = \sqrt{(6 - (-1))^2 + (-2 - 5)^2} = \sqrt{(6 + 1)^2 + (-2 - 5)^2} = \sqrt{7^2 + (-7)^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2}
\]

### Step 2: Find the perimeter
The perimeter of the triangle is the sum of the lengths of its sides:
\[
\text{Perimeter} = EF + FG + EG = 7 + 7 + 7\sqrt{2} = 14 + 7\sqrt{2}
\]

### Step 3: Find the area
Since this is a right triangle, the area can be found using the formula for the area of a right triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, \( EF = 7 \) and \( FG = 7 \) act as the base and height, respectively:
\[
\text{Area} = \frac{1}{2} \times 7 \times 7 = \frac{1}{2} \times 49 = 24.5
\]

### Final Answers:
- **Perimeter:** \( 14 + 7\sqrt{2} \)
- **Area:** \( 24.5 \) square units

Would you like more details or have any questions about this?

Here are some related questions for further exploration:
1. How do you generalize the area formula for any triangle using its vertices?
2. Can you calculate the centroid of the triangle given its vertices?
3. How do you find the length of a diagonal in a rectangle or other polygons?
4. How does the distance formula change in 3D space?
5. What is the significance of a right triangle in coordinate geometry?

**Tip:** For any triangle, knowing two perpendicular sides simplifies the area calculation, but the distance formula is a general tool for any side.

Find the perimeter and area of the polygon with the given vertices: E(6, −2), F(6, 5), G(−1, 5).

Learn how to find the perimeter and area of a right triangle formed by the vertices E(6, −2), F(6, 5), and G(−1, 5). This problem involves the distance formula, Pythagorean theorem, and basic geometry concepts. Suitable for students in Grades 9-10.

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