We are tasked with finding the perimeter of triangle $\Delta VWX$, where the coordinates of the vertices are:

• $V(-4, 8)$
• $W(-4, 2)$
• $X(10, 0)$

To find the perimeter, we need to calculate the lengths of the three sides of the triangle: $VW$, $WX$, and $VX$.

Step 1: Calculate the length of $VW$

The points $V(-4, 8)$ and $W(-4, 2)$ have the same $x$-coordinate, so the length of $VW$ is the vertical distance between them, which is the difference in their $y$-coordinates: $VW = |8 - 2| = 6$

Step 2: Calculate the length of $WX$

To find the distance between points $W(-4, 2)$ and $X(10, 0)$, we use the distance formula: $WX = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substituting the coordinates: $WX = \sqrt{(10 - (-4))^2 + (0 - 2)^2} = \sqrt{(10 + 4)^2 + (-2)^2} = \sqrt{14^2 + (-2)^2} = \sqrt{196 + 4} = \sqrt{200} \approx 14.1$

Step 3: Calculate the length of $VX$

Similarly, for the distance between points $V(-4, 8)$ and $X(10, 0)$, we again use the distance formula: $VX = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substituting the coordinates: $VX = \sqrt{(10 - (-4))^2 + (0 - 8)^2} = \sqrt{(10 + 4)^2 + (-8)^2} = \sqrt{14^2 + 8^2} = \sqrt{196 + 64} = \sqrt{260} \approx 16.1$

Step 4: Calculate the perimeter

Now, the perimeter $P$ is the sum of the three side lengths: $P = VW + WX + VX = 6 + 14.1 + 16.1 = 36.2$

Thus, the perimeter of $\Delta VWX$ is approximately 36.2 units.

Would you like more details or have any questions?

Here are five related questions to deepen your understanding:

1. How is the distance formula derived from the Pythagorean theorem?
2. What happens if you have a triangle on a 3D coordinate plane? How would you calculate the perimeter?
3. Can you calculate the area of $\Delta VWX$ using these points?
4. How would you find the slope of each side of the triangle?
5. What is the significance of using absolute values when calculating vertical or horizontal distances?

Tip: For any right triangle, you can also use the Pythagorean theorem directly for sides at right angles.