Question

reflections rules!
try each rule. click the button for each choice.
select one of them and explain how the algebra of the rule connects to the reflection of the triangle.
○ $(x,y) \to (-x,y)$
○ $(x,y) \to (x,-y)$

Explanation:

Step1: Choose a reflection rule

Let's choose the rule \((x,y) \to (-x,y)\). This rule represents a reflection over the \(y\)-axis.

Step2: Identify triangle vertices

First, find the coordinates of the triangle's vertices. Let's assume the blue point is \((2,5)\), the red point is \((3,3)\), and the black point is \((5,6)\) (by counting grid squares).

Step3: Apply the reflection rule

Apply the rule \((x,y) \to (-x,y)\) to each vertex:

• For \((2,5)\): \((-2,5)\)
• For \((3,3)\): \((-3,3)\)
• For \((5,6)\): \((-5,6)\)

Step4: Connect algebra to reflection

The rule changes the \(x\)-coordinate's sign (keeping \(y\) the same), which mirrors the point across the \(y\)-axis (where \(x = 0\)). So each vertex's distance from the \(y\)-axis (horizontal distance, \(|x|\)) remains the same, but it's on the opposite side of the \(y\)-axis. This algebraic change (negating \(x\)) geometrically reflects the triangle over the \(y\)-axis.

(If we chose \((x,y) \to (x,-y)\), it's a reflection over the \(x\)-axis: \(y\)-coordinate sign flips, mirroring over \(x = 0\), keeping horizontal distance same, flipping vertical position.)

Answer:

For rule \((x,y) \to (-x,y)\) (reflection over \(y\)-axis): Algebraically, \(x\)-coordinate is negated ( \(y\) stays). Geometrically, this mirrors each vertex across the \(y\)-axis, so the triangle reflects over the \(y\)-axis, with each point’s horizontal distance from \(y\)-axis preserved ( \(|x|\) same) but side (left/right of \(y\)-axis) flipped.