Integrated Math 2: Mastering Similarity Transformations for Exam Success

📐 Similarity Transformations: Your Key to Exam Success! 🌟

Similarity transformations are fundamental in Integrated Math 2. They involve changing the size or position of a figure without altering its shape. Let's break down the key transformations:

1. Dilation 🔍

Dilation is a transformation that changes the size of a figure. It requires a center point and a scale factor. * If the scale factor is greater than 1, the figure enlarges. * If the scale factor is between 0 and 1, the figure shrinks. Example: Consider a triangle $ABC$ with vertices $A(1, 1)$, $B(2, 1)$, and $C(1, 2)$. Dilate this triangle with a scale factor of 2 from the origin. The new vertices $A'$, $B'$, and $C'$ are: $A'(2, 2)$, $B'(4, 2)$, and $C'(2, 4)$.

# Python code to perform dilation
def dilation(point, scale_factor):
    x, y = point
    return (x * scale_factor, y * scale_factor)

A = (1, 1)
B = (2, 1)
C = (1, 2)

scale_factor = 2

A_prime = dilation(A, scale_factor)
B_prime = dilation(B, scale_factor)
C_prime = dilation(C, scale_factor)

print(f"A' = {A_prime}, B' = {B_prime}, C' = {C_prime}")

2. Reflection mirror 🪞

Reflection is a transformation that flips a figure over a line, called the line of reflection. * Reflection over the x-axis: $(x, y) \rightarrow (x, -y)$ * Reflection over the y-axis: $(x, y) \rightarrow (-x, y)$ Example: Reflect the point $(3, 2)$ over the x-axis. The new point is $(3, -2)$.

# Python code to perform reflection over x-axis
def reflect_x(point):
    x, y = point
    return (x, -y)

P = (3, 2)
P_prime = reflect_x(P)
print(f"P' = {P_prime}")

3. Rotation 🔄

Rotation is a transformation that turns a figure about a fixed point, called the center of rotation. * Rotation of 90° counterclockwise: $(x, y) \rightarrow (-y, x)$ * Rotation of 180°: $(x, y) \rightarrow (-x, -y)$ * Rotation of 270° counterclockwise: $(x, y) \rightarrow (y, -x)$ Example: Rotate the point $(1, 4)$ by 90° counterclockwise about the origin. The new point is $(-4, 1)$.

# Python code to perform 90-degree counterclockwise rotation
def rotate_90(point):
    x, y = point
    return (-y, x)

P = (1, 4)
P_prime = rotate_90(P)
print(f"P' = {P_prime}")

4. Translation ⬆️➡️⬇️⬅️

Translation is a transformation that slides a figure without changing its orientation. * Translation by $(a, b)$: $(x, y) \rightarrow (x + a, y + b)$ Example: Translate the point $(2, 3)$ by $(1, -2)$. The new point is $(3, 1)$.

# Python code to perform translation
def translate(point, vector):
    x, y = point
    a, b = vector
    return (x + a, y + b)

P = (2, 3)
vector = (1, -2)
P_prime = translate(P, vector)
print(f"P' = {P_prime}")

Practice Problems ✍️

1. Dilate triangle $DEF$ with vertices $D(0, 0)$, $E(3, 0)$, and $F(0, 3)$ by a scale factor of 0.5 from the origin. Find the new vertices.
2. Reflect the point $(-2, 5)$ over the y-axis. What are the coordinates of the reflected point?
3. Rotate the point $(4, -1)$ by 180° about the origin. Find the new coordinates.
4. Translate the point $(-3, -2)$ by the vector $(4, 5)$. What are the coordinates of the translated point?

Answers

1. $D'(0, 0)$, $E'(1.5, 0)$, $F'(0, 1.5)$
2. $(2, 5)$
3. $(-4, 1)$
4. $(1, 3)$