Let's start by plotting the points $F(8, 1)$, $G(2, 1)$, $H(-2, -8)$, and $I(4, -8)$ to form the parallelogram $FGHI$. After plotting, we'll calculate the perimeter of the parallelogram.

Step 1: Plot the Points

• F = (8, 1)
• G = (2, 1)
• H = (-2, -8)
• I = (4, -8)

Step 2: Verify the Parallelogram

For $FGHI$ to be a parallelogram, opposite sides must be parallel and equal in length. We can confirm this by checking the slopes and lengths of the sides $FG$, $GH$, $HI$, and $IF$.

Step 3: Calculate the Length of the Sides

1. Length of $FG$: $FG = \sqrt{(8 - 2)^2 + (1 - 1)^2} = \sqrt{6^2 + 0^2} = \sqrt{36} = 6$

2. Length of $GH$: $GH = \sqrt{(2 - (-2))^2 + (1 - (-8))^2} = \sqrt{4^2 + 9^2} = \sqrt{16 + 81} = \sqrt{97} \approx 9.8$

3. Length of $HI$: $HI = \sqrt{(4 - (-2))^2 + (-8 - (-8))^2} = \sqrt{6^2 + 0^2} = \sqrt{36} = 6$

4. Length of $IF$: $IF = \sqrt{(4 - 8)^2 + (-8 - 1)^2} = \sqrt{(-4)^2 + (-9)^2} = \sqrt{16 + 81} = \sqrt{97} \approx 9.8$

Step 4: Calculate the Perimeter

The perimeter $P$ of parallelogram $FGHI$ is given by the sum of the lengths of all its sides: $P = FG + GH + HI + IF = 6 + 9.8 + 6 + 9.8 = 31.6$

Final Answer

The perimeter of the parallelogram is 31.6 units.

Would you like further details or have any questions?

Relative Questions:

1. How do you determine if four points form a parallelogram?
2. Can you explain how to find the midpoint of a line segment?
3. What is the significance of the slope in determining parallel lines?
4. How would you calculate the area of this parallelogram?
5. What are the properties of a parallelogram regarding diagonals?

Tip:

Always double-check the slopes of opposite sides to ensure they are equal when verifying if a quadrilateral is a parallelogram.