import matplotlib.pyplot as plt

Original vertices of ΔABC

A = (3, 0) B = (4, 1) C = (5, -1)

Reflection function over the line y = 1

def reflect_over_y_equals_1(point): x, y = point return (x, 2 - y)

Reflect vertices of ΔABC

A_reflected = reflect_over_y_equals_1(A) B_reflected = reflect_over_y_equals_1(B) C_reflected = reflect_over_y_equals_1(C)

Plot the original triangle

plt.figure(figsize=(8, 8)) plt.plot([A[0], B[0], C[0], A[0]], [A[1], B[1], C[1], A[1]], label="Original ΔABC", marker='o')

Plot the reflected triangle

plt.plot([A_reflected[0], B_reflected[0], C_reflected[0], A_reflected[0]], [A_reflected[1], B_reflected[1], C_reflected[1], A_reflected[1]], label="Reflected ΔA'B'C'", marker='o')

Plot the line of reflection y = 1

plt.axhline(y=1, color='gray', linestyle='--', label='y = 1')

Labels and legend

plt.title("ΔABC and its Reflection over the Line y = 1") plt.xlabel("x-axis") plt.ylabel("y-axis") 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.legend() plt.gca().set_aspect('equal', adjustable='box')

Show the graph

plt.show() <<ImageDisplayed>>successHere is the graph of triangle ΔABC with vertices A(3, 0), B(4, 1), and C(5, -1), along with its reflected image across the line $y = 1$. The dashed line represents the line of reflection.

Would you like further clarification or a breakdown of the reflection process?

Here are 5 related questions to expand your understanding:

1. How is the reflection of a point calculated algebraically?
2. Can you reflect the triangle over a different line, such as $x = 2$?
3. How would the coordinates change if the triangle were reflected over the $x$-axis?
4. What is the general formula for reflecting a point across an arbitrary horizontal line $y = k$?
5. Can you compute the distances between corresponding points of the original and reflected triangles?

Tip: When reflecting over $y = k$, keep the $x$-coordinate constant and adjust the $y$-coordinate symmetrically based on $k$.