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, minus, 7, comma, 3, right parenthesis, and , left parenthesis, minus, 3, comma, minus, 4, right parenthesis

(−7,3) and (−3,−4)




To solve this problem, we will:

1. **Graph the right triangle formed by the two given points** \( (-7, 3) \) and \( (-3, -4) \).
2. **Calculate the distance between the two points** (the hypotenuse of the triangle) using the distance formula:
   \[
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]
3. **Find the lengths of the legs** of the triangle, representing the horizontal and vertical distances between the points.

---

### Step 1: Calculate the Horizontal and Vertical Distances
- The horizontal distance between the points is:
  \[
  |x_2 - x_1| = |-3 - (-7)| = |-3 + 7| = 4
  \]

- The vertical distance between the points is:
  \[
  |y_2 - y_1| = |-4 - 3| = |-4 - 3| = 7
  \]

---

### Step 2: Use the Distance Formula
The distance between the points is:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the values:
\[
d = \sqrt{(4)^2 + (7)^2} = \sqrt{16 + 49} = \sqrt{65}
\]

To the nearest tenth:
\[
d \approx 8.1
\]

---

### Step 3: Graph the Right Triangle
The right triangle is formed by:
- One leg along the horizontal axis from \( (-7, 3) \) to \( (-3, 3) \) (length = 4).
- Another leg along the vertical axis from \( (-3, 3) \) to \( (-3, -4) \) (length = 7).
- The hypotenuse connecting \( (-7, 3) \) and \( (-3, -4) \).

Let me generate the graph. matplotlib.pyplot as plt

# Points
x1, y1 = -7, 3
x2, y2 = -3, -4

# Plot the points
plt.figure(figsize=(8, 8))
plt.plot([x1, x2], [y1, y2], 'r-', label='Hypotenuse')
plt.plot([x1, x2], [y1, y1], 'b--', label='Horizontal Leg')
plt.plot([x2, x2], [y1, y2], 'g--', label='Vertical Leg')
plt.scatter([x1, x2], [y1, y2], color='black', zorder=5)

# Annotate points
plt.text(x1 - 0.5, y1 + 0.5, f"(-7, 3)", fontsize=10, color="black")
plt.text(x2 - 0.5, y2 - 0.5, f"(-3, -4)", fontsize=10, color="black")

# Set grid, labels, and aspect ratio
plt.axhline(0, color='black',linewidth=0.5)
plt.axvline(0, color='black',linewidth=0.5)
plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5)
plt.gca().set_aspect('equal', adjustable='box')
plt.legend()
plt.title("Graph of the Right Triangle")
plt.xlabel("x-axis")
plt.ylabel("y-axis")

plt.show()The graph displays the right triangle formed by the two points \( (-7, 3) \) and \( (-3, -4) \). The hypotenuse (red line) represents the distance between the points, while the horizontal (blue dashed) and vertical (green dashed) lines are the legs of the triangle.

### Final Results:
- **Horizontal leg**: \( 4 \) units
- **Vertical leg**: \( 7 \) units
- **Distance (hypotenuse)**: \( \sqrt{65} \approx 8.1 \) units

Would you like further details or have any additional questions? 

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### Related Questions:
1. How do you find the midpoint of two points?
2. Can you explain why the triangle formed is always a right triangle?
3. How would the calculations change if one of the points was at the origin?
4. How can you apply the Pythagorean theorem to verify the distance?
5. What are the properties of the slope of the hypotenuse in this example?

### Tip:
Remember, the distance formula is derived from the Pythagorean theorem! Always double-check by calculating the squares of the legs.

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). (-7,3) and (-3,-4).

Learn how to graph a right triangle and calculate the distance between points (-7,3) and (-3,-4) using the distance formula. Explore the step-by-step process of finding the horizontal and vertical distances, applying the Pythagorean theorem, and determining the hypotenuse to the nearest tenth. Ideal for Grades 7-10.

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