I DO (Concept Explanation)

Let's define the fundamental dimensions of coordinate space:

• Point: A location in space with 0 dimensions (width, height, or depth). Represented as $(x, y)$.
• Line: A straight, continuous, 1-dimensional path extending infinitely in opposite directions.
• Plane: A flat, 2-dimensional surface extending infinitely in all directions.

Line Relationships & Slopes: The slope ($m$) of a line measures its steepness: $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Corresponding Angles: When two parallel lines are crossed by a third line (called a transversal), the angles in matching corners are equal (congruent). Think of them as slide-shifted copies of each other!

WE DO (Guided Exploration)

Let's look at two lines on a coordinate map:
Line 1: Passes through points $(1, 3)$ and $(3, 7)$.
Line 2: Passes through points $(-2, 4)$ and $(0, 8)$.

Let's find the slope of Line 1: $$m_1 = \frac{7 - 3}{3 - 1} = \frac{4}{2} = 2$$ Now, let's find the slope of Line 2: $$m_2 = \frac{8 - 4}{0 - (-2)} = \frac{4}{2} = 2$$ Since $m_1 = m_2 = 2$, these lines are parallel! If we draw a horizontal line across both of them, the angles formed where they cross will be identical (corresponding angles).

YOU DO (Your Turn!)

Find your graph paper. Plot Line A using coordinates $(1, 2)$ and $(4, 8)$.

1. Calculate the slope of Line A.
2. Find the slope of a line that would be perpendicular to Line A.
3. Write down the slope of a line that is parallel to Line A.

I DO (Concept Explanation)

A circle is defined as all points in a 2D plane that are a fixed distance (the radius, $r$) from a central point $(h, k)$.

• Diameter ($d$): The distance across the circle through its center ($d = 2r$).
• Circumference ($C$): The distance around the circle ($C = 2\pi r$ or $C = \pi d$).
• Area ($A$): The total space enclosed inside the circle ($A = \pi r^2$).

Equation of a Circle: $(x - h)^2 + (y - k)^2 = r^2$
Where $(h, k)$ is the center coordinate, and $r$ is the radius.

Translating a Circle: Moving a circle is incredibly simple because its shape doesn't stretch or warp. To translate a circle, we just slide its center point $(h, k)$ to a new coordinate, leaving the radius unchanged.

WE DO (Guided Exploration)

Let's analyze a circular energy barrier in a video game with the equation: $$(x - 3)^2 + (y - 2)^2 = 16$$

• Where is the center? Since the standard form has $(x - h)$ and $(y - k)$, the center $(h, k)$ is at $(3, 2)$.
• What is the radius? $r^2 = 16$, so $r = \sqrt{16} = 4$ units.
• Area: $A = \pi r^2 = \pi (4)^2 = 16\pi \approx 50.27$ square units.
• Circumference: $C = 2\pi r = 2\pi (4) = 8\pi \approx 25.13$ units.

Now, let's translate (slide) this barrier 5 units left and 3 units up.
Our original center is $(3, 2)$.
New x-coordinate: $3 - 5 = -2$
New y-coordinate: $2 + 3 = 5$
The new center is $(-2, 5)$. Since the size of the barrier didn't change (rigid motion), the radius is still 4.
The new equation is: $$(x - (-2))^2 + (y - 5)^2 = 16 \implies (x + 2)^2 + (y - 5)^2 = 16$$

YOU DO (Your Turn!)

Use your compass or trace a circular object on your coordinate paper to draw a circle with a center at $(0, 0)$ and a radius of 3 units.

1. Write the algebraic equation for this starting circle.
2. Translate your circle 4 units to the right and 2 units down. Draw the new circle on your paper.
3. Write the new equation of your translated circle.
4. Calculate the exact area of your circle using $\pi$.

I DO (Concept Explanation)

Rigid Motion: A transformation that changes the position or orientation of a shape but does not alter its size or shape. The pre-image (original) and the image (transformed shape) are perfectly congruent.

There are three main types of rigid transformations:

• Translation (Slide): Slid in any direction. Code rule: $(x, y) \rightarrow (x + a, y + b)$.
• Reflection (Flip): Mirrored over a line (like the x-axis or y-axis).
  Rule for flipping over x-axis: $(x, y) \rightarrow (x, -y)$.
  Rule for flipping over y-axis: $(x, y) \rightarrow (-x, y)$.
• Rotation (Turn): Rotated around a fixed point (usually the origin, $(0,0)$).
  90° Clockwise rule: $(x, y) \rightarrow (y, -x)$.

WE DO (Guided Exploration)

Let's take a game character's triangular avatar with vertices:
$A(1, 1)$, $B(4, 1)$, and $C(1, 5)$.

Let's apply two back-to-back transformations to simulate gameplay physics:

1. First: Reflect over the y-axis (simulating character turning around to run left).
   Applying rule $(x, y) \rightarrow (-x, y)$:
   $A'( -1, 1)$, $B'( -4, 1)$, $C'( -1, 5)$
2. Second: Translate 3 units down (falling off a platform).
   Applying rule $(x, y) \rightarrow (x, y - 3)$:
   $A''( -1, -2)$, $B''( -4, -2)$, $C''( -1, 2)$

If we measure the side lengths of triangle $A''B''C''$, they are identical to the starting triangle $ABC$. Rigid motion preserved!

YOU DO (Your Turn!)

On your graph paper, draw a simple "L" shape using these four coordinates:
$P(2, 2)$, $Q(4, 2)$, $R(4, 3)$, and $S(2, 5)$.

1. Translate your shape 5 units to the left and 1 unit up. Label the new points $P'$, $Q'$, $R'$, and $S'$.
2. Take your new shape and reflect it over the x-axis. Write down the final coordinates of this new image.