To solve this problem and graph the right triangle:

Step 1: Plot the two points

The points are:

• $(6, -2)$
• $(4, 6)$

These form the endpoints of the hypotenuse of the right triangle.

Step 2: Draw the right triangle

The horizontal and vertical lines forming the legs of the right triangle can be obtained by:

1. Drawing a vertical line from $(4, 6)$ to $(4, -2)$, parallel to the $y$-axis.
2. Drawing a horizontal line from $(4, -2)$ to $(6, -2)$, parallel to the $x$-axis.

This forms the right triangle, with one leg along the x-axis and the other along the y-axis.

Step 3: Use the distance formula

To calculate the distance between the two points, we apply the distance formula:

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

Substituting the values:

$d = \sqrt{(6 - 4)^2 + (-2 - 6)^2} = \sqrt{(2)^2 + (-8)^2} = \sqrt{4 + 64} = \sqrt{68}$

$d \approx 8.2$

Final Answer:

The distance between the two points is approximately 8.2 units.

Would you like any further details or have any questions?

Here are 5 related questions:

1. What is the significance of the Pythagorean Theorem in calculating distances between points?
2. How can you determine the slope of the hypotenuse from the given points?
3. Can the midpoint formula help in other types of geometric problems?
4. How does the distance formula relate to vector operations in higher dimensions?
5. How would the process change if you were working in 3D coordinates?

Tip: In coordinate geometry, right triangles often help visualize distances and relationships between points, so it’s useful to understand how to construct them using perpendicular lines.