Graph a right triangle with the two points forming the hypotenuse. Using the sides, find the distance between the two points, *to the nearest tenth* (if necessary).

left parenthesis, 6, comma, 4, right parenthesis, and , left parenthesis, minus, 1, comma, minus, 2, right parenthesis

(6,4) and (−1,−2)

Let's go through the steps to graph the right triangle and find the distance between the two points \((6, 4)\) and \((-1, -2)\), rounding the distance to the nearest tenth.

### Step 1: Identify the Points
- Point A: \((6, 4)\)
- Point B: \((-1, -2)\)

### 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:
\[
L_1 = |x_2 - x_1| = |-1 - 6| = |-7| = 7
\]

#### Leg 2 (Vertical Distance, \(L_2\)):
This is the difference in the y-coordinates of the two points:
\[
L_2 = |y_2 - y_1| = |-2 - 4| = |-6| = 6
\]

### Step 3: Calculate the Hypotenuse (Distance Between the Points)
The hypotenuse is the distance between the points \((6, 4)\) and \((-1, -2)\), which can be calculated using the distance formula:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the values:
\[
\text{Distance} = \sqrt{(-1 - 6)^2 + (-2 - 4)^2} = \sqrt{(-7)^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85} \approx 9.2
\]

### Final Answer:
- **Leg 1 (Horizontal):** \(7\)
- **Leg 2 (Vertical):** \(6\)
- **Distance (Hypotenuse):** \(9.2\) (rounded to the nearest tenth)

Would you like to see the graph of the triangle or have further explanations? Here are some related questions you might find interesting:

1. What is the slope of the hypotenuse between the points \((6, 4)\) and \((-1, -2)\)?
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. What is the area of the triangle formed by these points?
6. Can the distance formula be used for three-dimensional coordinates?
7. How do you determine if a triangle is a right triangle using just the coordinates?
8. How does changing one of the points affect the distance between them?

**Tip:** The distance formula is a direct application of the Pythagorean Theorem in a coordinate plane, making it a powerful tool for finding the distance between any two points.

Learn how to graph a right triangle with points (6, 4) and (-1, -2), calculate the distance between these points using the distance formula, and round the result to the nearest tenth. This problem covers coordinate geometry, the distance formula, and the Pythagorean theorem. Suitable for students in Grades 9-12.

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