To graph the right triangle and find the distance between the two points $(3, -5)$ and $(9, 1)$, we can proceed with the following steps:

Step 1: Identify the Points

• Point A: $(3, -5)$
• Point B: $(9, 1)$

Step 2: Calculate the Lengths of the Legs of the Triangle

The legs of the right triangle are parallel to the x-axis and y-axis.

Leg 1 (Horizontal Distance, $L_1$):

This is the difference in the x-coordinates of the two points: $L_1 = |x_2 - x_1| = |9 - 3| = 6$

Leg 2 (Vertical Distance, $L_2$):

This is the difference in the y-coordinates of the two points: $L_2 = |y_2 - y_1| = |1 - (-5)| = |1 + 5| = 6$

Step 3: Calculate the Hypotenuse (Distance Between the Points)

The hypotenuse is the distance between the points $(3, -5)$ and $(9, 1)$, which can be calculated using the distance formula: $\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substituting the values: $\text{Distance} = \sqrt{(9 - 3)^2 + (1 - (-5))^2} = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} \approx 8.5$

Final Answer:

• Leg 1 (Horizontal): $6$
• Leg 2 (Vertical): $6$
• Distance (Hypotenuse): $8.5$

Would you like a detailed explanation of any part of this, or do you have any further questions? Here are some related questions to explore:

1. What is the slope of the hypotenuse between the points $(3, -5)$ and $(9, 1)$?
2. How would you calculate the midpoint of the hypotenuse?
3. What are the coordinates of the third vertex of the right triangle?
4. How does the Pythagorean Theorem apply in this scenario?
5. Can the distance formula be used for three-dimensional coordinates?
6. How do you determine if a triangle is a right triangle using just the coordinates?
7. What is the area of the triangle formed by these points?
8. How does changing one of the points affect the distance between them?

Tip: Remember that the Pythagorean Theorem only applies to right triangles, where the square of the hypotenuse equals the sum of the squares of the other two sides.